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HI BOOK 52 1 1.L3 1 c. 1 LAPLACE » TREATISE OF COLESTIAL MECHANICS 3 T1S3 DDl^TbbD 1 A TREATISE OF CELESTIAL MECHANICS, BY P. S. LAPLACE, MEMBER OF THE NATIONAL INSTITUTE, AND OF THE COMMITTEE OF LONGITUDE, OF FRANCE ; THE ROYAL SOCIETIES OF LONDON AND GOTTINGEN; of THE ACADEMIES OF SCIENCES OF RUSSIA, DENMARK, AND PRUSSIA, &C. PART THE FIRST— BOOK THE FIRST. TRANSLATED FROM THE FRENCH, AND ELUCIDATED WITH EXPLANATORY NOTES. BY THE REV. HENRY H. HARTE, F.T.C.D. M.R.LA. DUBLIN: PRINTED FOR RICHARD MILLIKEN, BOOKSELLER TO THE UNIVERSITY ; AND FOR LONGMAN, HURST, REES, ORME AND BROWNE, LONDON. 1822. O. CBAISBBBRY, FRniTEB TO IHE VNrrERSITT. TO THE REV. CHARLES WILLIAM WALL, THIS TREATISE IS DEDICATED, BY HIS FRIEND, AND FORMER PUPIL, HENRY H. HARTE, A 3 PREFACE. It has been made a matter of surprise, that considering the great capabilities of many individuals in these countries, so few are conversant with the contents of a work of such acknowledged eminence, as the Celestial Mechanics. Without adverting to other causes, it may be safely asserted, that the chief obstacle to a more general knowledge of the work, arises from the sum- mary manner in which the Author passes over the intermediate steps in several of his most interesting investigations. To re- move this obstacle, is the design of the present treatise, in which the translator endeavours to elucidate every diffi- culty in the text, and to expand the different operations which are taken for granted. He has not attempted to follow the principles into all their details; but he has occasionally adverted to some useful applications of them, which occur in different Authors. He is aware that those conversant with such subjects will find much observation that may be dispensed with ; but when it is considered that his object was to render this work accessible to the general class of readers, he trusts that he will not be deemed unnecessarily diffiise, if he has insisted longer on some points than the experienced reader may think neces- sary. As many of the propositions which Newton announced se- parately are so many different results, which are all comprised VI PREFACE. under the same general law analytically investigated, he has also taken occasion to notice, in the notes, those propo- sitions of Newton, which are embraced in the general analysis of the text, which he was induced to do, in order to show the great superiority of the analytic mode of investigating problems. The Work will be divided into five parts, which will be published in separate volumes. The first volume contains the first book, which treats of the general prin- ciples of the equilibrium and motion of bodies. The number of notes which was necessary for the elucidation of these prin- ciples is much greater than will be required in any of the subsequent volumes. The second volume will contain the second and third books of the original ; the third volume, the fourth and fifth books ; the fourth volume will contain the sixth, seventh, and eighth books ; and the last volume will contain the ninth and tenth books, together with the supplement to the tenth book. Trin. Coll. April, 18S2, TABLE OF CONTENTS. BOOK I. Of the generallaws of Equilibriwn, and of Motion. .... page 1 CHAP. I, Of the Equilibrium and of the composition of Forces which act on a material point. ............ ibid. Of Motion, of Force, of tlie Composition and Resolution of Forces. - Nos. 1 and 2 Equation of the Equilibrium of a point soUicited by any number of Forces acting in any direction. Method of determining, when the point is not free, the pressure which it exerts on the surface, or against the curve, on which it is constrained to exist. Theory of Moments. .......... No. 3 CHAP. II. Of the Motion of a Material Point. 21 Of the law of inertia, of uniform motion and of velocity. ... No. 4 Investigation of the relation which exists between the force and the velocity ; in the case of nature they are proportional to each other. Consequences of this law. Nos. 5 and 6 Equations of the motion of a point sollicited by any forces whatever. - No. 7 General expression of the square of its velocity. It describes a curve in which the integral of the product of its velocity, by the element of this curve, is a minimum. No. 8 Method of determining the pressure, which a point moving on a surface, or on a curve, exerts against it. Of the centrifugal force. ...... No. 9 Application of the preceding principles to the motion of a free point actuated by the force of gravity, in a resisting medium. Investigation of the law of resistance necessary to make the point describe a given curve. Particular examination of the case in wliich the resistance vanishes. ......... No. 10 Application of the same principles, to the motion of a heavy body on a spherical surface. Determination of the duration of the oscillations of the moving body. The very small oscillations are isochronous. ..-..-- No. 11 VUl CONTENTS. Investigation of the curve, on which the isochronisra obtains rigorously, in a resisting medium, and particularly, when the resistance is proportional to the two first powers of the velocity. -...-.,.,.. No. 12 CHAP. III. Of the Equilibrium of a system of bodies. .... "75 Conditions of the equilibrium of a system of points wliich impinge on each other, with directly contraiy velocities. What is understood by the quantity of motion of a body, and by similar material points. --..... No. 13 Of tlie reciprocal action of material points. Reaction is always equal and contrary to action. Equation of the equilibrium of a S3-stem of bodies, fi-om which results the principle of virtual velocities. Method of determining the pressures, which bodies exert on the curve cr on the surface on which they are subjected to move. . - , No. 14 Application of these principles, to the case in which all the points of the system are in- variably connected together ; conditions of the equilibrium, of such a system. Of the centre of gravity. Method of determining its position; 1st, with respect to three fixed and rectangular planes ; 2diy, with respect to three points given in space. - No. 15 Conditions of the Equilibrium of a solid body of any figure whatever. - No. 16 CHAP. IV. Of the Equilibrium 0/ fluids. ...... 99 General equations of tliis equilibrium. Application to the equilibrium of a homogenous fluid mass, of which the exterior surface is free, and which is spread over a fixed solid nucleus, of any f gure whatever. ....... No. 17 CHAP. V. The general principles of the motion of a system of hodiei, - 108 General equation of this motion. ....... No. 18 Developement of the principles which it contains. Of the principle of living forces. It only obtains, when the motions of the bodies change by insensible gradations. Method of estimating the change which the living force sustains, in the sudden variations of the motion of the system. ......... No. 19 Of the principle of the conservation of the motion of the centre of gravity ; it subsists even in the case in which the bodies of the system exercise on each other a finite action in an instant. No. 20 Of the principle of the conservation of areas. It subsists in like manner as the preceding principle, in the case of a sudden change in the motion of the system. Determination of the system of coordinates, in which the sum of the areas described by the projections of the radii vectores vanishes for tw6 of the rectangular planes formed by the axes of the coordinates. This sum is a maximum for the third rectangular plane ; it vanishes for every other plane perpendicular to this last. ... No. 21 Tlie principles of the conservation of living forces and of areas, obtain also, if the origin of the coordinates, be supposed to have an uniform and rectilinear motion in space. In this case, the plane passing constantly through this point, and on which the sum of the CONTENTS, IX areas described by the projections of the radii is a matimum, remains always parallel to itself. The principles of living forces and of areas may be reduced to certain relations between the coordinates of the mutual distances of the bodies of the system. The planes passing through each body of the system, parallel to the invariable plane drawn through the centre of gravity, possess analogous properties. .... No. 22 Principle of the least action. Combined with the principle of living forces, it gives the general equation of motion. ........ No. 23 CHAP. VI. Of the laws of motion of a system bodies, in all relations mathematically possible between the force and the velocity. ...... 152 New Principles which, in this general case, correspond to those of the conservation of living forces, of areas, of the motion of the centre of gravity, and of the least action. In a system which is not acted on by extraneous forces, 1 °, the sum of the finite forces of the system, resolved parallel to any axis, is constant ; 2° the sum of the finite forces to make the system revolve about an axis is constant ; 3° the sum of the integrals of the finite forces of the system, multiplied respectively by the elements of their directions, is a minimum s these three sums vanish in the case of equilibrium. - - No. 24 CHAP. VII. Of the motion of a solid body of any figure xjohateverm - • 159 Equations which determine the motion of translation, and of rotation of a body. Nos. 25, 26 Of the principal axes. In general a body has only one system of principal axes. Of the moments of inertia. The greatest and the least of these moments appertain to the prin- cipal axes, and the least of all the moments of inertia belongs to one of the three prin- cipal axes which passes through the centre of gravity. Case in which the solid has an infinite number of principal axes. -.....- No. 27 Investigation of the instantaneous axis of rotation of a body : the quantities which de- termine its position relatively to the principal axes, determine at the same time the velocity of rotation. ........... No. 28 Equations which determine in a function of the time, this position and that of the principal axes. Application to the case in wliich the motion of rotation arises fi-om an impulsion which does not pass through the centre of gravity. Formula for determining the distance of this centre from the direction of the initial impulsion. Example deduced from the planets, and particulariy from the earth. ...... No. 29 Of the oscillations of a body which turns very nearly about one of its principal axes. The motion is stable, about the principal axes of which the moments of inertia are the greatest and the least ; it is not so about the third principal axis. - • No. 30 Of the motion of a solid body about a fixed axis. Determination of the simple pendulum which oscillates in the same time as the body. ..... No. 31 CHAP. VIII. Of the motion offuids. 222 Equations of the motion of fluids ; condition relative to their continuity. - No. 32 b X CONTENTS. Transformation of these equations ; they are integrable, when the density being any function o!" the pressure, the sum of the velocities parallel to the rectangular coordinates, re- spectively multiplied by the element of its direction, is an exact variation. It is demon- strated, that if this condition obtains at any one instant, it will always obtain. No. 35 Application of the preceding principles to the motion of an homogeneous fluid mass, which revolves uniformly about one of the axes of the coordinates. ... No. 34 Determination of the very small oscillations of an homogeneous fluid mass, which covers a spheroid having a motion of rotation. ...... No. 35 Applicatior. to the motion of the sea, on the supposition that it is deranged from the state of equilibrium by the action of very small forces. . . - . No. 36 Of the terrestrial atmosphere considered at first in a state of equilibrium. Of the oscillation* which it experiences in the state of motion, and considering only the regular causes whicli iigitate it ; of the variations which those motions produce in the heights of the barometer. ........,-- No. 37 A TREATISE OB CELESTIAL MECHANICS, &c. &c, JN EWTON published, towards the close of the seventeenth century, the discovery of universal gravitation. Since that period. Philosophers have reduced all the known phenomena of the system of the world to this great law of nature, and have thus succeeded in giving to the theories and astronomical tables a precision which could never have been anticipated. I propose in this present treatise to exhibit in one point of view, these theories which are scattered through a great number of works, of which the whole comprising the results of universal gravi- tation, on the equilibrium and motion of the bodies both solid and fluid, composing the solar and similar systems, constitutes The Celestial Mechanics. Astronomy, considered in the most general manner, is a great problem of Mechanics, of which the arbitrary quantities are the elements of the motions of the heavenly bodies j its solution depends, at the same time, on the precision of the observations, and on the perfection of analysis ; and it is of the last importance to banish all empiricism, and to reduce it, so that it may borrow nothing from observation, but the indispensable data. The object of this work, is, as far as it is in my power, to accomplish this interesting end. I trust that, in consideration of the difficulties and importance of the ^ xii subject, Philosophers and Mathematicians will receive it with indulgence, and that they will find the results sufficiently simple to be employed in their investigations. It will be divided into two parts. In the first, I will give the methods, and formulas, for determining the motions of the centres of gravity of the heavenly bodies, their figures, the oscillations of the fluids which are spread over them, and their motions about their proper centres of gravity. In the second part, I will apply the formulse which have been found in the first, to the planets, the satellites and the comets -, and I will conclude with a discussion of several questions relative to the system of the world, and by a historical notice of the labours of Mathematicians on this subject. I will adopt the decimal division of the quadrant, and of the day, and I will refer the linear measures, to the length of the metre, determined by the arc of the ter- restrial meridian comprised between Dunkirk and Barcelona. ERRATA. Page Line 3, 23, Jbr the new forces, read these forces. 6, 12, Jbr reluctant, read resultant. 12, 15, for {c) read {b). 17, i>Jor equation, read equations. 23, U,forq>{f),read <i>'CfJ- 2i, 14i, for a and b, read c and b ; and line 17, for angle, read jangled. 32, lOj^or di/*dz^, read dy- -^dz-. 33, ' 2, fiom bottom, after A^, add — ; and last line, after the differential of the, add square of the, and for s''ds, read . 35, 2, from bottom, for first the order, read the first order. 40, 18, after centrifugal, add force. 47, last line, for dt constant, read dx constant. 49, ll,yb>- 2/i. COS. 6. read Ih. cos. -i. ; and in lines 21, 22, dele the 2 which occur in the Den". 50, 14, dele the 2 by which the values oidt, dz, dx, are multiplied. 51, 1\, for git, read gf^. 65, \,fordsi,readds'%;\m&lQ,for\o^n.{s-\-q) — ), readXog. 7i.{s-\-q)) — ; h'ne 17, for{s'-\-q^') reads'-\^. 82, 4, for P, read —P. 83, 2, for they, read it. 84, 23, /or i, k, k, read R, R, R. 86, 3 from boitom, for -^ read -r— . ox OX 94, 2, a/ier centre, rearf of gravity. 99, 12, for figure, rearf figures. 105, for Sg, read Sp ; 20, after each, rwrf other. ERRATA. Page Line 138, 16, Jbr sin. C. sin. -J'.+cos. ■^. sin. <p. read sin. 6. sin. -J^. cos. •4'. sin. (p ; line 19, for makes read make. 142, IQyforl.mx, "Zmy, ^mz, read Smx,, Xjmj/^, 5?»i2,. 148, 18, /or 2»<. (22)n./m. Prfx+ Q,dy-\-Rdz), read 2»i. 22.fm,(Pdx+ Qdy+Rdz and in line 19, for Jrtim'^ fdf, readJinm'.Fdf. 149, 2, <j/?pr velocities, reac? of the bodies. 168, 15, to the second t-'- prefix + and line 19, /b?- + x" read + x"'. 197, 20, /or -f r', read -f /". 204, 21, _/or parallel, read perpendicular. 21 4, 2, for to coincide very nearly with the plane of x' and j/ ', rertt;? to be very nearly perpendicular to the plane of x and j/'. 217, 26, for dy, read dy . 218, 19, for 2^» sin. 6. readz"^. sin. ^6. 229, 2, for '^^, read ^. do dc 230, 3, /or f/j, reflf/ dr, line 11, /or the first ^.^, read 4 --t,- '' db da da do du „ , , fdiu 231, 20, /or ^. V + read i-^\i. dz ^dxj j.dt dx nao A ^ du.+dtv.dt , du + dv.i 233, 4,/or-:2_]Z ^ read -^ 234, 2, multiply the first member by t?t, line 11, prefix — toc?i,and/)r — V read — P'.dt. 235, 17, /or kread^. k 236, 17, /)r Jr, rend S^ and for homogenous, read homogeneous. 240j. 16, for dz, read dz^ . 241, 12, for the s^Qond— — -, read-r-. dx^ dz* 251, 16, for ra^, read r^ a. 252, 8,/ornumbers, ?earf members. 256, 17, /or r». (sin. «-f-a!<. cos.«); read r'fsin. e-\-au cos. 6) ; line 19,for2as,read2ars. 265, 17, /or -, rcac?.^. TREATISE ON CELESTIAL MECHAJVICS, S(c. Sfc. PART I.— BOOK I. In this book, the general principles of the equilibrium and motion of bodies are established, and those problems in Mechanics are solved, the solution of which is indispensable in the theoiy of the system of ^le world. CHAPTER I. Of the equilibrium and of the composition of forces which act on a material point. 1. A body appears to us to move, when it changes its situation with respect to a system of bodies which we suppose to be at rest; but as all bodies, even those which seem to us to be in a state of the 2 CELESTIAL MECHANICS, most absolute rest, may be in motion ; we, in imagination, refer the position of bodies to a space which is supposed to be boundless, im- moveable, and penetrable to matter ; and when they answer succes- sively to diflPerent parts of this real or ideal space, we conceive them to be in motion. The nature of that singular modification, in consequence of which a body is transferred from one place to another is, and always will be, un- known : we have designated it by the name force ; but we can only detennine its effects and the laws of its action. The effect of a force acting on a material point, is, if no obstacle opposes, to put it in mo- tion ; the direction of tlie force is the right line which it tends to make the point describe. It is evident that when two forces act in the same direction, their effect to move the point is the sum of the two forces, and that when they act in opposite directions, the point is moved by a force represented by their difference. If their directions form an angle with each other, a force results, the direction of which is intermediate between the directions of the composing forces. Let us investigate this resultant and its direction. For this purpose, let us consider two forces :c and J/ acting at the same time on the material point AI, and forming a right angle with each other. Let z represent their resultant, and 0 the angle which it makes witli the direction of the force x ; the two forces ^ and 1/ being given, the angle 6 will be determined, and also the resultant z, so that there exists between these three quantities j:, t/, z, a relation which it is required to ascertain. Let us suppose at first the forces x and 1/ infinitely small, and equal to the differentials dJ! and dy ; let us suppose afterwards that a: becom- ing successively dx, Q.dx, Sdx, &c. y becomes dy, '2dy, Sdy, &c. it is evident that the angle 9 will always remain the same, and that the resultant a; will becone successively rf^, 2dz,^3dz, &c. ; thus in the successive incremen's of tlie three forces x, y, and z, the ratio ol x to « will be constat, ^ and can be expressed by a function of fl which we will desig!iate by ^(6) ; therefore we shall have x = z 9(9), in PART I.— BOOK I. 3 which equation x may be changed into y, provided that in like manner the angle 0 is changed into — 8, w being the semi-circumference of a circle whose radius is equal to unity. Now, we can consider the force x as the resultant of two forces 'x' and J*, of which the first ^'"is in the direction of the resultant z, the second x' being perpendicular to this resultant. The force x which results from these two new forces, forming the angle 6 with the force J?', and the angle— — 9, with the force x" we shall have therefore we can substitute these two forces, for the force x. In like manner we can substitute for the force y, two new forces y' and y\ of which the first is equal to — and in the direction of 2;, and of which the second is equal to li and perpendicular to z, thus we shall have in place of the two forces x and y the four following, x* y"^ xy xy z z z z the two last acting in opposite directions, destroy each other ; the two first acting in the same direction, when added together constitute the resultant z ; we shall have therefore x»+3/» — jt. from which it follows that the resultant of the two forces x and y n represented in quantity by the diagonal, of a rectangle, of which the sides represent the new forces. Let us now proceed to determine the angle 6. If the force x is B2 4 CELESTIAL MECHANICS, increased by its differential, without altering the force y,* this angle will be diminished by the indefinitely small quantity J9, but it is pos- sible to suppose the force dx resolved into two, one dx' in the direc- tion of s, the other dx'' perpendicular to z; the point Mwill then be acted on by the forces z + dx' and dx' perpendicular to each other, and the resultant of those two forces, which we represent by z\ will make with dx" the angle ~ — rffl ; therefore by what precedes we shall have dx" = z'. f^— — d&j, consequently the function (pfZ d^'\ is indefinitely small, and of the form — Kd^ ; K being a constant quantity independent of the angle 6 ; therefore we have dx" =1 — Kd^ ; z' differing by an indefinitely small quantity from z ; moreover as dx" forms an angle with dx equal to — — 0 we have dx" = dx <p( 9 j = 1/. dx ; therefore rf6 = — ydx Kz* * Since the direction of the resultant depends on the relation which exists between composing forces, if one force be increased, while the other remains unaltered, the angle contained between the direction of the increased force and resultant, will be diminished by a quantity of the same order with that by which the force was increased. And when the force y receives the increase, the angle contained between the resultant and this increased force, will be diminished, therefore its complement, the angle i, will be increased by the same quantity ; and this is the reason why the expressions for the variations of 6 corresponding to the respective variations of x and y are affected with contrary signs. If X and y are increased or diminished simultaneously, d6 will always vanish when dx, dy are respectively proportional to the quantities varied ; this follows immediately from th« expression for di. PART L— BOOK I. 5 If the force y is varied by its differential dy, x being supposed to be constant, we shall have the corresponding variation of the angle 6, by changing x into y, y into x, and fl into — —9, in the preceding equa- tion ; which then gives xdy therefore by making x and y to vary at the same time, the total va- riation of the angle 6 will be -^ JL .and we shall have xdy—ydx _ ^^^ If we substitute for x;* its value .r* +^', and then* integrate we shall have -^ - tan. (ii:fi + p) X f being a constant arbitrary quantity. This equation being combined with the equation j^'+j/' =2^ gives vT = 2. cos. (iiCS+p) * xd\i — y dx ^ xdv — y dx j / U \ du 1 +^ 1 +£_ I» x^ , J—= u \ therefore f—J^!— { - arc tang. = « ) rzf K di = \ X J COS. Kl+( 6 CELESTIAL MECHANICS. It is only now required to know the two constant quantities A" and p ; but if we suppose y to vanish we have evidently z = a\ and ^ = o, therefore cos. /> = 1 and x -zz z. cos. K^. If we suppose x to vanish, then z ■=. y, and 9 — — w ; cos. K^ being then equal to nothing, K *must be equal to 2«4-l, n being an integral number; and in this case .r will vanish as often as 9 will be equal to ^^ ; but x being no- thing we have evidently 9 zz A-ct ; therefore 2«+l zz 1, or n zz o, consequently X = z. COS. 9. From which it follows that the diagonal of a rectangle described on the right lines which represent the forces x and y, represents not only, the quantity but also the direction of their reluctant. Thus we can substitute for any force whatever two other forces which form the sides of a rectangle, of which that force is the diagonal ; and it is easy to infer from thence that it is possible to resolve a force into three others, which form the sides of a rectangular parallelipiped of which it is the diagonal. t Let therefore a b and c represent the three rectangular coordi- nates of the extremity of a right line, which represents any force what- ever, and of which the origin is that of the coordinates ; this force will be represented by the function s/a*-\-h''' -\-c*, and by resolving it * In this case K6 is some odd multiple of -■— and therefore K must be of the form 2n-|- 1 . f Tlie given force being resolved into two, of which one is perpendicular to a plane given in position, the other being parallel to tliis plane, if this second partial force be decom- posed into two others, parallel to two axes situated in this plane, and perpendicular to each other ; it is evident that the three partial forces will be at right angles to each other, and that the sum of the squares of the lines representing these forces, will be equal to the square of the line representing the given force, therefore this last force is the diago- nal of a rectangular parallulUpiped, of which the partial forces constitute the sides. PART I.— BOOK I. 7 parallel to the axes of a o£ b and of c, the partial forces will be ex pressed respectively by these coordinates. Let a', b', &, be the coordinates of a second force ; a-\-a', b-\-b', c+c', will be the coordinates of the resultant of the two forces, and will represent the partial forces into which it can be resolved parallel to the three axes, from whence it is easy to conclude that this resultant is the diagonal of a parallelogram, of which the two forces are the sides.* In general a, b, c ; a', b>, C ; a", b«, &' ; &c. being the coordinates of any number of forces ; a -{■ a' ■{■ a" -^ , &c. b+b'+b''+, kc.c-\-c'-\-c"-\- &c. will be the coordinates of the resultant ; the square of which will be equal to the sum of the squares of these last coordinates ; thus we shall have both the quantity and the position of the resultant. t * The coordinates of the extremity of this diagonal are evidently equal to n+a', h-\-b', c+c, therefore tliis diagonal must be equal to the resultant of the two forces. We are enabled to derive an expression for the cosine of the angle, contained between the given forces, in terms of the cosines of the angles which these forces make with the coordi- nates, for calling the forces S and S', and the angles which S makes with the three axes, A, A', A", and B, B\ B", the angles which S' makes with the same axes we have o=Scos. A, b=S COS. A, c=S cos. A",a'=-S cos.B,c'=S cos. 5', c' = S' cos. B' ; the square of the line connecting the extremities of S and S = S* — Q,SS. cos. il+.S ' ; ^ being the angle contained between S and S, the square of this line is also equal to (S cos. A—S cos. BY + {S cos. A'—S' cos. £')=+ (S cos. A'—S cos. B')* ; =» S' + S"^— 2 SS' (cos. A. cos. B + cos. A. cos. 5'4-cos. A', cos. B",) consequently we have cos. A = cos. A. cos. B + cos. A . cos. B + cos. A", cos. B', therefore when the two forces are perpendicular to each other, the second member of this equation is equal to nothing. t Let S S' S", &c. represent the forces of wliich the coordinates are respectively a, 6, c ; a, V , c' ; a", b", c", &c. then by what precedes a-\-a', b-\-b', c+c', are the co- ordinates of the resultant of S and S', a+a'+a", b-\-b'-\-b", c+c'+c', .are the coor- dinates of the resultant of this last force, and the force S" &c. : therefore the resultant f of any number of forces is the diagonal of a rectangular parallelipiped of which 8 CELESTIAL MECHANICS, 2. From any point whatever of the direction of a force S, which point we will take for the origin of this force, let us draw a right line, which we will call 5, to the material point M ; let x, y, z, be the three rectangular coordinates which determine the position of the point M, and a, b, c, the coordinates of the origin of the force ; we shall have If we resolve the force S parallel to the axes of .«■, of i/, and of z, the corresponding partial forces will be by the preceding number S r S ' S ' \izJ ySl/J WS^ the coordinates are equal respectively to the sum of the coordinates of the composing forces, V F»= (a+a'+a" +&c.y- + (6+*'+*" &c.)* + {c+c +c' + &c.)'. Let m,n, p = the angles which V makes with the rectangular axes a4-fl'+o"+ &'C h + b' + b"+&c. c+c' + c" + &c. COS. m = ^ y— cos. m = ^ ^^JE cos. p = _L_X,_L — •. • we have both the quantity and direction of the resultant. From the preceding composition of forces it follows, that if a polygon is constructed, of which the sides, (which may be in different planes) are respectively proportional to these forces, and parallel to their directions, the last side of this polygon represents the resultant of all the forces in quantity and in direction. * S being considered as a function of x, y, and k, S« ^ ( T~ J ^''''^ { s" J ^■^''' ( T~} ^^ and when s = V{x-ay+(y-by+(z-c-' [jj = -j- j^ = '-^ ' jr =-^ ' &c. are evidently the expressions for the cosines of the angles which s s s makes with the coordinates x, y, and s, since PART I— BOOK I. 9 ■ >;> -i, expressing according to the received notation Ws ^ _ { is tlie coefficients of the variations of Sx, Sy, Sz, in tlie variation of the preceding expression of s. If, in like manner, we name s' the distance of M from any point iu the direction of another force iS', that point being taken for the origin of this force ; S'. \——l will be this force resolved parallel to the axes I SxS of .r, and just so the rest ; therefore the sum of the forces S, S', S", kc. V '■■{ |) = M-^) + s-(t; ) + *'(t^) + '"■ by iiuiltiplying these equations by Sx, 3y, h, respectively, and adding them together, we get >'■'«= K(^).''+(|>'.+(t:)'-0 +.S'Y (l!l)ix+/!^^ 3y ■)-( ^)y Sz+&c.=Sls-t-.S'?.v' + .S'3,v"+ &c.=-Z.S.h. Now since these equation have phice whatever be the variations 3x, Si/; h, one of then; may exist while the other two vanish, therefore the equation (a) is equivalent to the tliree Kiuations which precede it. We shall see liereafter that the introductio)i of the coeffi- cient ( Y^ ) is of the greatest consequence, for from the equation (4) which tbllows immediately from the equation (a), we deduce the equation (/) of No. li, which involves the principle of vertual velocities, and this principle combined with that of D'Alembert, lias given to Mechanics all the perfection of which it was susceptible, for by means of it tTie investigation of the motions of any system of bodies is reduced to the integration of differential equations. .See No. 18. 10 CELESTIAL MECHANICS, resolved parallel to this axis will be 2. S.( — ), the characteristic 2: of (is \ ( Ss' ) Sx ( Sxi Let F be the resultant of all the forces S, S', &c. and u the distance of the point M from any point in the direction of this resultant, which is taken for its origin ; V. < > will be the expression of this re- ' aX J sultant resolved parallel to the axis of x; therefore by the precedhig number we shall have V.< — '-> = H. S.< ( Sx ) l we shall have in like manner Ss_ \ Sy ^ ^ SyS ' Ws ^ ^ cT^ ♦ from which we may obtain, by multiplying these equations respectively by to, Sy, Sz, and adding them together VJu = 1. S. is; As this last equation has place whatever be the variations Sx, Sy, Sz it is equivalent to the three preceding. If its second member is an exact variation of a fuction <p, we shall have F. Su = S({>, and consequently Stp which indicates that the sum of all the forces resolved parallel to the axis of ■>■- is equal to the partial difference ) — ^ i . * This case ob- * If we mult;ply h the variation of any quantity by any function of that quantity, such ^ as -a — ' ^.s™, &c. the product is evidently an exact variation, however this is not true of every species of function, for there are some transcendental and exponential functions. such as which are not exact variations. log. .?. PART L— BOOK I. U tains generally, when the forces are respectively functions oF the dis- tance of their origin from the point M. In order to have the resultant of all these forces resolved parallel to any right line whatever, we shall take the integral S. /." S. is, and naming <p this integral, we shall consi- der it as a function of .v, and of two right lines perpendicular to each other and to x ; the partial difference < — ^ > will be the resultant I Sx ^ of the forces S S' S", &c. resolved parallel to the right line x. 3. When the point AI is in equilibrio, in consequence of the action of the forces which solicit it ; their resultant vanishes, and the equa- tion (a) becomes O = ■£. S. Ss {b) which indicates, that in the case of the equilibriiun of a point acted on by any number of forces, the sum of the products of each force by the element of its direction is nothing.* c2 * Since the forces parallel to the coordinates .c, y, z, are independant of each other, It follows from the notes to the preceding number, that M'hen the point M is in equilibrio -• -S -J -— J- 2. S. -! —1 !• 2. i>. I __ I. are = respectively to nothing. t. c. ,S. cos. A^S. COS. B-^S" COS. C+ etc. = 0 S. COS. A'-\-S' cbs. 5'+S" cos. C'+&c. = 0. a. COS. A''-\-&'. COS. £"+ S'' COS. C" = 0. {A, A', A" ; B, B', B", &c. are the angles which the direction* of ,S', S>, &c.niake with J-, y, z,) ; these are the equations of equilibrium of a system of forces applied to a mate- rial point which is entirely free. The independence which exists between these equations is ejttremely advantageous, it only obtains \\hen the forces are resoh'ed paralle! to three rectangular coordinates. 2. S. ) -z— > =0 indicates that M is at an invuriable dis- t Sx 3 tance from the plane of y, z ; in this case the forces are reducible to two rectangular ones, in the plane y, z. When the point M is in equilibrio any one of the forces acting on it is equal ami contrary to the resultant of all the remaining forces, for naming V the resultant of the forces .S', S"-)-itc. and n, h, c, the angles which it makes with the coordmates x, y, z, by 12 CELESTIAL MECHANICS, If the point M is forced to be on a curved surface, it will experi- ence a reaction, which we will designate by R. This reaction is equal and directly contrary to the pressure with which the point presses on the surface ; for by conceiving it acted on by two forces R and — R, it is possible to suppose that the force R is destroyed by the reaction of the surface, and that thus the point presses the surface witli the force R ; but the force of pressure of a point on a surface is perpen- dicular to it, otherwise it might be resolved into two, one perpendicular to the surface, which would be destroyed by it, the other parallel to the surface, in consequence of which the point would have no action on this surface, which is contrary to the hypothesis ; consequently if r be tlie perpendicular drawn from the point ^/ to the surface, and termi- nated in any point whate\''er of its direction, the force R will be di- rected along this perpendicular ; therefore it will be necessary to add R.Sr to the second member of the equation (c) which thus becomes O = Z, S. Ss-{-RAr {c) — R being then the resultant of all the forces .V, S', &c. it is perpen- dicular to the surface. If we suppose that the arbitrary variations Sx, Sij, Sz belong to the cui'ved surface on which the point is subjected to remain, we shall have h- — O, since r is perpendicular to the surface, therefore RJr vanishes from the preceding equation, in consequence of which the equation (b) obtains in this case, provided that one of the three variations Sx, St/, Sx, be eliminated by means of the equation to the surface ; but then, tht- what precedes we shall have V'. cos a = S' cos. B-\-S" cos. C^&c. /'' cos. c = S' COS. B'-fS". COS. C'-\-&c. and since S. cos. A+S. cos. B+S". cos. C-f&c. zz 0. We have v. cos. a= — 6'. cos. A ; in like manner it may be shewn that f" cos. b= — S. cos. B, and v. COS. c = — S. cos. C ; if we add together the squares of these equations we shall obtain F'"=S*, because cos. *a -f cos. *i -|- 'c = 1 = cos. -4+ cos. 'B 4- cos. *C.-. we have cos. a = — cos. A &c. •.• a = 200' — A, iu like manner it follows, that b = 20O — B, c = 200— C, v the forces S and V are equal, and act in opposite directions. PART I.— BOOK I. 13 equation (b) which in the general case is equivalent to three, is only equivalent to two distinct equations, which may be obtained by putting the coefficients of the two remaining differentials separately equal to nothing. Let m = 0 be the equation of the surface, the two equations Sr—0, and SuzzO will have place at the same time ; this requires that h- should be equal to \Su, N being a function of x, i/, and z. Naming a, b, c, the coordinates of the origin of r we shall have to determine it from which wc may obtaui -^ — > + < — , + { j — I, and consequently therefore by making I X :r R ^ (- Sx ^ ^ Sy ^ ^ Sz ^ the term R.Sr of the equation (c) will be changed into xiu, and this equation will become 0 r: I. iS'. h-irxiii in which equation we ought to put the coefficients of the variations Sx, ii/, iz, separately equal to nothing, which gives three equations ; but on ac- count of the indeterminate quantity a, which they contain, they are equi- valent to only two between x, y, and a. Therefore instead of extracting from the equation {hi) one of the variations Sx, Sy, Sz, by means of the differential equation of the surface, we may add to it this equation multi- plied by the indeterminate quantity a, and then consider the variations Sx, %, and Sz, as independant. This method, which also results from 14 CELESTIAL MECHANICS, the tlieory of elimination combines the advantage of simplifying the calculation with that of indicating the force — R with which the point il/ presses the surface.* * When the point M is on a curved surface, then all that is required for its equilibrium is, that the direction of the resultant of all the forces which act on it should be perpen- dicular to this surface, but the intensity of this resultant is altogether undetermined, since the reaction is equal and contrary to the pressure of the point on the surface, by adding to 2. S. is the quantity R. Sr we may consider the material point as entirely tree. 3r vanishes because the perpendicular is the shortest line which can be drawn from a given point to the surface. Since the same values of x, y, and z, satisfy the equations 2r = 0 S« = 0, it follows from the theory of equations that 'N = is a function of .r, y, and z, "ill this function it follows from the expression that is given for Jr,' that the cosines of the angles wliit-h the noi-mal makes with the coordinates are equal respectively to iV. < " ' A'. > Jf i .V. j 1: ^ . iix' ^ »y ^ ^" See notes to No. 9. \%z S \^z S \ %z ) then 2. S. S.s -f y.%u=Ci will be equal to X. ix-\- Y. Jy+ 2. 5-- + and on account of the independance of the variables x, y, z, we shall have eliminating a we liave the following equations : y. ??i-x.i!i=o, Z. ^JL-X.h =0. 3x ly Ji ^~ PART I.— BOOK I. 15 Let us conceive this point to be contained in a canal of simple or double curvature ; the reaction of the canal which we will denote by k, will be equal and directly contrary to the pressure with which the point acts against the canal, the direction of which is perpendicular to its side ; but the curve formed by this canal, is the intersection of two sur- faces of which the equations express its nature, therefore we may con- sider the force k as the resultant of two forces R, R', which express the reactions of the two surfaces on the point M ; since the directions of the three forces li, R', /., being respectively perpendicular to the side of the curve they are in the same plane, therefore by naming h; Sr' the elements of the directions of the forces R, R', which directions are respectively perpendicular to each surface ; we must add to the equation (A) the two terms RSr, R'Sr, which, will change it into the following : Q-^.SSs + R.Sr-^R'.h'. (dj These are the equations of equilibrium of a material point solicited by any number of forces S, S, S', and constrained to move on a curved surface : if the position of M on the sur- face is not given, then the two equations, resulting from the elimination of a, combined with the equation of the surface, m:=0, are sufficient to determine the three coordinates of the point. Wlien the forces and position of the point are given we obtain a by means of one of the three preceding equations, from which we can collect immediately the value of R, and consequently the pressure ; the investigation of R would be considerably abridged f ^" 1 ( ^" I by making the axis of j- to coincide with the normal, for then a. < y- >• , x.-\ |r~ ( ' are equal respectively to nothing, and a -! -i— r = i? A'. 4 — > = /f , for in this case ""•Vi] = li] = ^= ^'"^'^ ^il'r ^{4}' '"^ = '" ""''^^' '^^ '•*'" have y = 0, Z = 0, which indicate that the forces resolved respectively parallel to two lines in the plane which touches the surface in the given point, are equal to nothing ; this also follows from considering that the resultant of the forces is necessarily perpendicular to the surface. If the variations 5.r, ly, Iz, are supposed to belong to the surface then we shall have XS-r-J- Y'^y\-Z'iz = 0, and substituting for Sz its value in terms of ?x and 5^, which we get by means of the equation \ -r- f • ^-^ + ^ "j" !" • ^^ + "! "sT ( ' ^^ ~ ^• we can obtain immediately the equations of condition ix dy dx iz 16 CELESTIAL MECHANICS, If we deterniiue the variations ix, Si/, Sz, so that they may appertain at the same time to the two surfaces, and consequently to the curve formed by the canal ; Sr and Sr' will vanish, and the preceding equation will be reduced to the equation (^b) which therefore obtains in the case where the point is constrained to move in a canal ; provided that we make two of the variations ix, Sy, Sz, to disappear by means of the two equations which express the nature of this canal. Let us suppose that u = 0, ?/— 0 are the equations of the tAvo surfaces whose intersection forms the canal. If we make R V(|) Su \- / hi \ * , , Su ^ Sx Sy oz '■ the equation (d) will become 0=2. S. Ss. + A. Su + x'.Su', in which the coefficients of each of the variations Sd; Sy, Sz, will be se- parately equal to nothing ; thus three equations will be obtained, by means of which the values of a and x' may be determined, which will give R and R' the reaction of the two surfaces, and by composing them we shall have the reaction k of the canal on the point AI, and conse- quently the pressure of this point against the canal. The reaction re- solved parallel to the axis of a: is equal to ^ Sx' ' ^ Sx ^ Sx ^ Sx •' * When the point is forced to be on a canal of simple or double curvature there is only one equation of condition, which is obtained by eliminating A and >' ; this equation combined with the equations m = 0, «' = 0 are sufficient to detmiine the coordinates of the PART L— BOOK I. 17 therefore the equation of condition u = 0, u'=0, to which the motion of the point M is subjected, express by means of the partial differentials of functions, which are equal to nothing in consequence of tliese equa- tions, the resistances with which the point is affected in consequence of the conditions of its motion. It appears from what precedes that the equation (/>) of equilibrium obtains universally, provided, that the variations Sj:, Sy^ Sz, are subjected to the conditions of equilibrium. This equation may be made the foun- dation of the following principle. " If an indefinitely small variation be made in the position of the " point M, so that it still remains on the curve or surface along which " it ought to move, if it is not entirely free ; the sum of tiie forces " which solicit it, each multiplied by the space through which the " point moves in its direction, is equal to nothing, in the case of an " equilibrium."* The variations Sx, Sy, iz, being supposed arbitrary and independant, it is possible to substitute for the coordinates .r, y, z, in the equation (a), three other quantities which are functions of them, and to equal the coefficients of the variations of these quantities to nothing. Thus naming p the radius drawn from the origin of the coordinates, to the D point of the canal where the given forces constitute an equilibrium, in this case it is only required for the equilibrium of the point that the resultant of the forces should exist in a plane perpendicular to the element of the curve on which the point is situated, from whence it appears that the position of the resultant is more undetermined than when the point exists on a curved surface. See Notes to No. 9. We might simplify the investigation of the pressures and obtain immediately the equation of equilibrium between ths forces by taking two of the axes in the plane of the normals of the surfaces whose intersection constitutes the curve, for then we shall have at once Z—O, the third axis is in the direction of the tangent to the curve formed by the intersection of the two given surfaces. * The equation (b) obtains universally, but under different circumstances, according at tlie point is free, or constrained to move on a surface ; in the former case V the resultan: of all the forces vanishes, and vS.S.Ji. r= V.hi must vanish; in the latter case Fhasfe 16 CELESTIAL MECHANICS, projection of the point M, on the plane of x and y, and it the angle formed by p and the axis of x, we shall have T=p. COS. TT ; yzzfi. sin. v. If, therefore in the equation [a), we consider 2^, s, sf as functions of - f. It, and 2 ; and then compare the coefficients of Si?, we shall have _1 S^ ^ is the expression for the force V resolved in the direction o( the element p. S-ar. Let V be this force resolved parallel to the plane of X and y, and P the perpendicular demitted from the axis of z on PV' direction of V', parallel to the same plane ; will be a second ex- P pression for the force V resolved in the direction of the element f Jw ; therefore we shall have PF.= V. Su \- If we conceive the force V to be applied to the extremity of the per- pendicular P, it will tend to make it turn about the axis of Z ; the product of this force, by the perpendicular, is denominated the moment of the force V with respect to the axis of z ; therefore this moment is equal to V.\ —1 ; and it appears from the equation (e), that the moment of the resultant of any number of forces is equal to the sum of the moments of these forces.* finite value, but its direction being perpendicular to the surface or the variation of this per- pendicular must be equal to nothing, and consequently in this case also ^.Sis^ Viu must vanish. * The force V resolved parallel to the axis of a = -^-^ =, by substituting for PART L— BOOK I. I9 X its vftlue V.^ '■ ) this last force resolved in the direction of the elemeni u f. dv, i. e. perpentBcuter to ^= V.— l^_i'-^ tr (by substituting for 1/ its value) u ^ V. iSl '. • sin. a- in like manner if we resolve the force V parallel to the axis of «, and then this last force in the direction of 5 3»-, it frill be equal to V. U'^'";""— ). ^j^g^ ^ ti These forces in the direction of j. Stt act in opposite directions, therefore their difference ^^ ((?■ sin. a- — I)). COS. T — (j. cos. ?r — a), sin. ?r)_) is the expression for that part of die force V in the direction of the element ^.Jjr, which is really efficient, this expression """■{■£■}' ^""^ "'" ^^' ^"^' '^— ")"+(?•*'"• ^—l>'+(z—cy (by substituting for J and 1/ their values) ; therefore taking the derivitive function, a- being considered as the variable, we shall have, u. i -r— f = — g. sin. a-. (5. cos. a- — 0)+^. cos. v. (5. sin. 3- 6y. ••• — \ ^\ = — U<^"^- '^- («• ^'°- 'r— 6J— sin. T. (?. COS. ^—a)),= for conceiv- g t OTT J U p ing the force V to be resolved into two, of which one is perpendicular to ^ , tlie otlier being in the direction of 5, the triangle constituted by tliese forces will be similar to a triangle, two of whose sides are ^ and P, and the third side = F' produced to meet P, ■■■ that part of the force V wliich is perpendicular to 5 is to V as P to 5 •.• it is equal to PV From the definition that has been given in this No. of the moment of a force with respect to an axis, it appears that it can be geometrically exliibited by means of a triangle, whose vertex is in this axis, and whose base represents the intensity of the , force, it vanishes when the resultant V vanishes, and also when P vanishes, i. e. when the resultant piisses through the origin of the coordinates. See Notes to No. 6. Let X and Vindicate, as in the preceding notes, the force V, resolved respectively pa- rallel to the axes of x and 3^, X=F.ifi:±l, Y^ V.^MlI^, the expression for these u u forces resolved perpendicular to e=F. i^^llii-- -, V.liUJ. f, their difference « J " f = = ^ ; we are enabled by means of tliis expression to deduce the equa- tion of the right line, along which the resultant is directed, for the equations of its pro- d2 20 CELESTIAL MECHANICS, jection V on the plane of x y is y— ^ = — — . ( x — a), Xy — Xh = Yx — Ylc. Let L be A. equal to Yx — Xy, and the preceding equation will become b = — .a " we might \ A derive similar expressions for the projection of V on the planes of r and 2, and y and z, from whence it is easy to collect the equation of the right line along which V is directed, — Y" indicates the distance of the origin of the coordinates from the intersection of V with the axis of y, and -^ indicates the distance of the origin of the coordiaates from the intersection of the resultant V with the axis of x. Yx — Xy= Ya — Xh shews that it is indifferent what point of the direction of V is considered. Yx — Xy = 0 when V = 0, and also when its direction passes through the axis o z. PART I— BOOK I. 21 CHAPTE21 II. Of the motion of a material point. 4. A point in repose cannot excite any motion in itself, because there is nothing in its nature to determine it to move in one direction in pre- ference to another. When solicited by any force, and tlien left to itself, it will move constantly, and uniformly in the direction of that force, if it meets with no resistance. This tendency of matter to persevere in its state of motion or rest, is what is termed its inertia ; it is the first law of the motion of bodies. The direction of the motion in a right line follows necessarily from this, that there is no reason why the point should deviate to the right, rather than to the left of its primitive direction ; but the uniformity of its motion is not equally evident. The nature of the moving force being unknown, it is impossible to know a priori, whether this force should continue without intermission or not. Indeed, as a body is in- capable of exciting any motion in itself, it seems equally incapable of producing any change in that which it has received, so that the law of inertia is at least the most natural and the most simple which can be imagined ; it is also confirmed by experience. In fact, we observe on the earth that the motions are perpetuated for a longer time, in pro- portion as the obstacles which oppose them are diminished ; which induces us to think that if these obstacles were entirely removed, the motions would never cease. But the inertia of matter is most remark- able in the motions of the heavenly bodies, which for a great number of ages have not experienced any perceptible alteration. For these rea- sons we shall consider the inertia of bodies as a law of nature ; and when we observe any change in the motion of a body we shall conclude that it arises from the action of some foreign cause. €2 CELESTIAL MECHANICS, In uniform motions the spaces described are proportional to the times. But the times employed in describing a given space are longer or shorter according to the magnitude of the moving force. From these differences has arisen the idea of velocity, which, in uniform motions is the ratio of the space to the time employed in describing it. Thus s representing the space, / the time, and v the velocity, we have v— — . Time and space being heterogeneal and consequently not comparable quantities, a determinate interval of time, such as a second, is taken for a unit of time, and in like manner a portion of space, such as a metre for an unit of space, and then time and space become abstract numbeis, which express how often they contain units of their species, and thus they may be compared one with another. By this means the velocity becomes the ratio of two abstract numbers, aiad its unity is the velocity of a body vi^hlch describes a metre in one second. 5. Force being only known to us by the space which it causes to be described in a given time, it is natural to take this space for its measure, but this supposes, that several forces acting in the same direction, would cause to be described in a second of time, a space equal to the sum of the spaces which each would have caused to be described separately in the same time, or in other words, that the force is proportional to the velocity ; but of this we cannot be assured a p7~iori, in consequence of our ignorance of the nature of the moving force. Therefore it is ne- cessary on this subject also to have recourse to experience, for whatever is not a necessary consequence of the few data which we have on tke nature of things, must be to us the result of observation. Let us name v the velocity of the earth, which is common to all the bodies on its surface, let f be the force with which one of these bodies. M is actuated in consequence of this velocity, and let us suppose that V ~ f'9{,fy is the relation which exists between the velocity and the force, ^f) being a function oi f which must be determined by expe- rience. Let a, b, c. be the three partial forces into which the force / may be resolved parallel to three axes which are perpendicular to each other. Let us then suppose the moving body M to be solicited by s PART L— BOOK I. 23 new force, f, which may be resolved into three others a', h', c, pa- rallel to the same axis. The forces by which this body will be soli- cited parallel to these axis will be a-\-a', b-\-b\ c-{-c', naming F the sole resulting force, by what precedes we shall have F = y^^'l « + (6+i,')« + (c-t-cO* If the velocity corresponding to i<'be named U ; * —■ — will be this velocity resolved parallel to the axes of a, thus the relative velo- city of the body on the earth parallel to this axis will be -^ — — — — '— or(a + «')' 'P'i.P) — <^' ff- The most considerable forces which can be impressed on bodies at the surface of the earth being much smaller than those by which they are actuated in consequence of the motion oi' the earth, we may consider «', 7/, c', as indefinitely small quantities relative to f; therefore we shall have F ^ f •\ „ t and ? (F) = <p. (/) 4- (««'+^^^+cc^ ^^j.^ . ^_^^,^ . j^^.^^ ^j^^ differential * The velocity of a body moving in a given direction is to its velocity, estimated in any other direction, as radius to the cosine of the angle which the two directions make with one another, that is, in this case as F to a+a', therefore the velocity U resolved parallel to the axis of a will be equal to U "l f F. = ^{a4-a')»-(-(i4-J')*4.(c-l.c')« = v/a'+5*+c^+2;(rt'+266'+2cc', the squares of a, b', and c being rejected as indefinitely small, if this radical is expanded by the binomial theorem (all the terms except the two first being neglected as involving the squares, products, and higher powers of a', V, c',) it will become \/a'+6*+c«+ 2 {aa'+bb'+cc) —f +aa'bb'+cc, 2 ya'+6»+^, / and (p (F) = ?.(/+ "1±^*I1^) equal by Taylor's tlieprem to 24 CELESTIAL MECHANICS, of (p.(f) divided by d.f. I'hc rcLitive velocity of M in the direction of the axis of a will thus beconie a-<^-(J') + 4 { ««' + ^''' + «' }• <?'- CfJ its relative velocities in the directions of b and c will be (/^Cf ) + y { .7f/ f- W/ + cc'l 9' (/) ; The position of the axes of a of ^ and of c being arbitrary, we may take the direction of the impressed force for the axis of a, and then b and c will vanish ; the preceding relative velocities will be changed into the following a I <?.{/) + ^ . p' C/) } .^. a'.<p' (/) ; -^- . </ <p' (/). If (p' [f) does not vanish, the moving body in consequence of the impressed force a' will have a relative velocity perpendicular to the direc- tion of this force, provided that a and b do not vanish,— that is to say, provided that the direction of this force does not coincide with that of the motion of the earth. Thus, conceiving that a globe at rest upon a very smooth horizontal plane is struck by the base of a right angle cy- linder, moving in the direction of its axis, which is supposed to be ho- rizontal, the apparent relative motion of the globe will not be parallel to this axis in all positions of this axis relative to the horizon. We have thus an easy means of determining by experiment whether ip'{J') has a perceptible value on the earth ; but the most accurate i xperiments have not indicated in the apparent motion of the globe any deviation from the direction of the force impressed ; from which it follows that on the earth <p'{f) is very nearly nothing. If its value was at all per- ceptible, it would particularly be shewn in the duration of the csciila- PART I.— BOOK I. 25 tions of a pendulum, which duration would alter according as the po- sition of the plane of its motion differed from the direction of the mo- tion of the earth. As the most exact observations have not evinced any such difference, we ought to conclude that <p'{J) is insensible, and at the surface of the earth ought to be supposed equal to nothing.* If the equation <p' (_/) = 0 has place whatever be the magnitude of the force J", ?>.(/') will be constant, and the velocity will be pro- portional to the force ; it will be also proportional to it if the function <?•{./) is composed of only one term, as otherwise ^'.(/") would not vanish unless J" did ; therefore if. the velocity is not proportional to the force, it is necessary to suppose that, in nature, the function of the velocity which expresses the force consists of several terms, which is very improbable ; we must moreover suppose that the velocity of the earth is exactly such as corresponds to the equation (pXX) ^^ ^'^ which is contrary to all probability. Besides, the velocity of the earth varies during the different seasons of the year ; it is a thirtieth part greater in winter than in summer. This variation is even more considerable if, as every thing appears to indicate, the solar system be in motion in space ; for according as this progressive motion conspires with that of the earth, or is contrary to it, there must result in the course of the year, very sensible variations in the absolute motion of the earth, which would alter the equation which we are considering, and the ratio of the force impressed to the absolute velocity which results from it, if this equa- tion and this ratio were not independant of the motion of the earth. Nevertheless, the smallest difference has not been discovered by observation. ■* These experiments evince that the appearances of bodies in motion are independant of the direction of the motion of the earth ; and from the preceding investigation it follows, that in order this should be the case, the small increase of the force by which the earth is actuated should be to the corresponding increase of the velocity, in the ratio of the quantities themselves; thus our experiments only prove the reality of this proportion, which if it had place, whatever the velocity of the earth might be, would give the law of the velocity proportional to the force. t <p' {/) = 0, not only when <p ( /) is constant, but also in other cases, such as when ip (y) is a maximum or minimum, in the former case the force _/ may be of any magnitude whatever ; in the latter case the value of y is unique ; but since the velocity S6 CELESTIAL MECHANICS, Thus we have two laws of motion ; the law of inertia, and that of the force proportional to the velocity, which are both given by observation. They are the most natural and the most simple which can be imagined, and are, without doubt, derived from the nature itself of matter, but this nature being unknown, they are, with respect to us, solely the re- sult of observation, and the only observed facts which the science of Mechanics borrows from experience.* 6. The velocity being proportional to the force, those two quantities may be represented one by the other, and we may apply to the compo- sition of velocities all that has been previously established respecting the composition of forces.t Thus it follows, that the relative motions of a system of bodies actuated by any force whatever, are the same whatever be their common motion, for this last motion decomposed into three othei-s, parallel to three fixed axes, only increases by the same quantity the partial velocities of each body parallel to these axes, and as their relative velocities only depend on the difference of these partial velocities, it will be the same whatever be the motion common to all bodies ; it is therefore impossible to judge of the absolute motion of the system, of which we make a part by the appearances which can be observed, which circumstance characterises the law of the force propor- tional to the velocity. of the earth is different in different points of its orbit, the value of / corresponding to this velocity must also vary. If (p (y) is an algebraic function ofyj and consists of only one term, then a' ( f) will not vanish unless y vanishes ; but if ip vcas a transcendental function, then / might have a finite value, tfiXJ') vanisihing, or vice versa, * In this respect, therefore, the theory of motion is less extensive than the theory of equilibrium, which does not involve any hypothesis whatever. f Let V, V, v", represent the uniform velocities parallel to the coordinates x, y, z, after any time t, x =-. v t, y = v'.f, z = v".t, tlie resulting motion will be uniform, and its di- rection rectilinear, the equation of,?, the line described, will be s =t ^v--i-v''-\-v''', the ve- locity in the direction of s = \/i;''-f-u' --J-u"^, the cosines of the angles which this di- rection makes with x, y, and z, are equal respectively to \/v' -Jfv'-+v"^, v/v^-f-v * + «'*, ^v'+v'^+v"' ; thus the composition and resolution of velocities are effected in the same manner as the composition and resolution of forces. PART I.— BOOK I. 27 It follows also from No. 3, that, if we project each force and their resultant on a lixed plane, the sum of the moments of the composing forces thus projected with respect to a fixed point taken on the plane, is equal to the moment of the projection of the resultant ; but if we draw from this point to the moving body a radius, which we shall call the radius vector, this radius projected on a fixed plane will trace, in con- sequence of each force acting separately, an area equal to the product of the projection of the line which the moving body is made to describe, into half the perpendicular let fall from the fixed point on this pro- jection ; therefore this area is proportional to the time ; it is also in a given time* proportional to the moment of the projection of the force ; thus the sum of the areas which the projection of the radius vector would describe, if each composing force acted separately, is equal to the area which the resultant makes this radius to describe. It follows from this, that if a body is first projected in a right line, and then solicited by any forces whatever, directed towards a fixed point, its radius vector will always describe about this point areas proportional to the times, because the areas which the new composing forcest make this radius to describe will vanish. It appears conversely, that if the moving body describes areas proportional to the times about the fixed point, the resultant of the new forces which solicit it is constantly directed towards this point.? E 2 * The area varies as the base muUiplied into the altitude ; the base varies as the time multiplied into the projection of the force ; therefore the area varies as the continued pro- duct of the altitude, projection of force, and time, or (by substituting the moment for the altitude multiplied into the projection of the force ) as the moment multiplied into the time. f If the forces directed to the fixed point did not act, the moving point would evi- dently describe areas proportional to the times ; but these forces being supposed to act, the areas which are describ'id about the fixed point, in consequence of the action of these forces, ai-e nothing ; for tlie perpendicular from the fixed point on the direction of the force in this case vanishes, consequently the proportionality of the areas to the times is not disturbed by the action of those forces. X By means of the equations, -r— ■ =P.- — ^ = Q. which are established in the sub- sequent number, we can exhibit immediately the relation which exists between the areas 28 CELESTIAL MECHANICS, 7. Let us now considei- the motion of a material point solicited by forces which seem to act continually, such as gravity. The causes of this and similar forces which have place in nature being unknown, it is im- possible to know whether they act without interruption, or whether their successive actions are separated by imperceptible intervals of time ; and moments ; for if we multiply the first of these equations by y, and the second by X, and then subtract, we shall have, by concinnating — — — — + yP — xQ) = 0; if this equation he integrated, we shall obtain — -^""-^ ■ -^ Jl dt (yP — xQ)=c; yP — xQ. is the moment of the projection of the force on the plane x and y (see last note to No. 3) ; it vanishes when the force is directed to the origin of the coordinates, and also when P and Q vanish, that is when the point is not solicited by any accelerating force, -consequently in both these cases, xdy — ydx = cdt and is •/ proportional to the time ; in the second case the origin of the coordinates may be any point whatever ; but in the first case, the origin must be in the^xerf point, to which the forces soliciting the point are directed; [xdy — ydx = the elementaiy area which the projection of tlie radius vector on the plane x y describes \r dt; for X = 5. cos. TT, y zz. ^. sin. 5r ; therefore dx =: d^. cos. 5r — d-n, sin. a-.g. dy^d^. sin. ■jr-\-d'!r. cos. 57.^. consequently xdy — y.dx ziz d^. sin. iv. cos. ;r.^-f-(/ir. cos.^jt.j- — d^. sin. ?r. COS. 3-.j+c?5r. sin. ^!r.§^=c?7r,g- ; but since ^dir is the elementary arc described by the projection of the radius vector on the plane x, y, g.-rfjrwill be the expression for the elementary area.) Since, wlien the areas are proportional to the times ^P — -rQrrO, it fol- lows that the magnitude of the area described m a given time is not affected by the in- tensity of the accelerating force. By a similar process of reasoning it may be shewn, that the projections of the elemen- tary area on the plane x, 2, y, 2, which are equal to xdz— zdx, ydz — zdy generally, are equal respectively to c'.dt, d'.dt. when the forces soliciting the point are directed towards the origin of the coordinates. When the areas are proportional to the times, the curve described is of single curvature ; for then we have xdy — ydx=cdt, xdz — zdxzuc'dt, ydz — zdy=id'dt ; if the first of these equations be multiplied by z, the second by y, and the third by x, we shall obtain, by adding them together, the equation cz-\-c'y'rcf'x = 0, which belongs to a plane. The velocities are inversly as the perpendiculars when the areas are proportional to the tmies ; for if we call the perpendicular p, and the elementary arc of the curve described ds, we will have p.ds = x.dy — y.dx = cdt •.• p = — = — . The constant quantities c, c, c', depend on the species of the curve described ; in conic sections when the force is directed to the focus, they are to the squaje roots of the para- meters as the cosines of inclinations of the planes x,y, x,z, yz, to the plane of the section to radius. See No. 3, book 2. PART I.— BOOK I. 29 but it is easy to be assured that the phenomena ought to be very nearly the same on the two hypotheses ; for if we represent the velocity of a body solicited by a force whose action is continued by the ordinate of a curve of which the abscissa represents the time, this curve, on the second hypothesis will be changed into a polygon, having a great number of sides, which for this reason may be confounded with the curve. We shall, with geometers, adopt the first hypothesis, and suppose that the interval between two consecutive actions is equal to the element dt of the time, which we will denote by t. It is evident that the action of a force ought to be more considerable accoi'ding as the interval is greater which separates its successive actions, in order that after the same time t the velocity may be always the same. Therefore the instantaneous action of a force ought to be supposed to be in the ratio of its intensity, and of the element of time during which it is supposed to act. Thus P, representing this intensity at the commencement of each instant, dt, the point, will be solicited by the force Pdt, and its motion will be uni- form during this instant. This being agreed upon. All the forces which solicit a point AI may be reduced to three, P, Q, R, acting parallel to three rectangular coordinates x, y, z, which determine the position of this point ;* we shall suppose these forces to act in a contrary direction from the origin of the coordinates, or to tend to increase them. At the commencement of a new instant dt, the moving point receives in the direction of each of its coordinates incre- ments of force or velocity, Fdt, Q.dt, Rdt. The velocities of the point M, parallel to these coordinates, are -1- ' -^> ^ ,t for during an inde- ^ dt dt dt *' * By thus referring the position of a point in space to rectangular coordinates, all curvilinear motion may be reduced to two or three rectilinear motions, according as the curve described is of simple or double curvature. For the position of the moving point is completely determined when we are able to assign the position of its projections on three rectangular axes, each coordinate represents the rectilinear space described by the point parallel to the axes to which it is referred, it will consequently be a given function of the time ; and if we could determine these functions with respect to the three coordinates, the species of the curve described might be assigned by eliminating the time by means of the three equations between the coordinates and the time. t The space being a function of the time, dx = v.dt is the limit of the value of the incre- 30 CELESTIAL MECHANICS, finitely small portion of time, they may be considered as uniform, and therefore eijual to the elementary spaces divided by the element of the time. Consequently the velocity with which the moving body is solicited at the connneii; enient of a new instant, is —+P.dt ; ^ +Q.di; Jl+Rdt; dt ' dt dt or ^+d.^-d.^+P.dt; dt dt dt ±+d.-^-d.^ + Q.dh dt dt dt ^ , ^A.d.J^^d. — +R.dt; dt^dt dt ^ but in this new instant, the velocities with which the moving body is actuated parallel to the coordinates x, y, z, are evidently dx , dx dii J dy dz , ^ dz 1- d. ; — ^ + d. -i^; +«• —z ; dt dt ' dt dt ' dt ^ dt' ment of the space, when dt becomes indefiiiitely small ; we can assign tJie actual value by means of Taylor's theorem ; for if i receive the increment dt, then {x=f{t) becomes x'=f{t^di) ^ , s dx , d-x df^ , d'x dl^ , ^ , ,. ... x'-x = / {i+dt)-f (t) = — . dt+--r- . _+--•—-+ &c. by malong dt' ^ df 1.2 ' dt^ 1.2.3 I sinrp ^ dt dx dt indefinitely small all the terms but the two first may be rejected ; and since -— is the d^x CoefiScient of dt it represents the velocity, and since is the coefficient of dt^, it is proportional to the force; consequently if the action of the forces solicit- d^t ing the point should cease suddenly —j^ would vanish, and the point would move d 'jc d^x with an uniform velocity, if instead of vanishing -— • became constant, then — — , and all subsequent coefficients would vanish, and the motion of the point would be composed of an uniform motion, and of one uniformly accelerated, both commencing at the same instant. PART I.— BOOK I. ' SI therefore the forces -d.— + V.dt, — d -^+ Ci-dt, -d. ^+B.dt, dt - dt dt must be destroyed, so that, if the point was actuated by these sole forces it would be in equilibriuiu. Thus if we denote by So:, St/, Sz, any varia- tions whatever of the three coordinates jt, t/, z, which variations are not necessarily the same with the differentials d^, dy, dz, that express the spaces described by the moving body parallel to the three coordi- nates during the instant dt, the equation {b) of No. 3, will become 0=,^;^. \d. ——P.dl.l -\-Sy, \d. !k—Q.dt.l +Sz.\d. ^—R.dtX. (/)* i dt i i dt 3 L dt ) We may put the coefficients of ^.r, Sy, Sz, separately equal to nothing ; if the point M be free, and the element dt of the time being supposed constant, the differential equations will become dt ■ ' dt" dt» * From the equation (J~) it appears that the laws of the motion of a material point may be reduced to those of their equilibrium, we shall see in No. 18, that the laws of the mo- tion of any system of bodies are reducible to the laws of their equiUbrium. f If P, Q, R, are given in functions of the coordinates, then by integrating twice we shall obtain the values of x, y, and z, in functions of the time ; two constant quantities are introduced by these integrations, the first depends on the velocity of the point at a given instant, the second depends on the position of the point at the same instant. If the values of the coordinates x, y, z, which are determined by these integrations, give equations of this form, x=a.f{t), yz:^b.f (t), z=c. fit), the point will move in a right line, the cosines of the angles which the direction of this line makes with x, y, and z, are respectively equal to — — . . — . the constant quantities «, b, c, depend' on the nature of the function y(0. if/(0 ='; a, b, c, re- present the uniform velocities parallel to x, y, and z, the uniform velocity of the point = \/a*+F+cs" if/(0 =<S then a, b, c, are proportional to the accelerating forces parallel to .r, y, z, aiul the point will be moved with a motion uniformly accelerated, repre- 32 CELESTIAL MECHANICS, If the point M be not free, but subjected to move on a curve or on ii surface, then by means of the equations to the curve or surface, there must be eliminated from the equation (f) as many of the variations Sx, Sy, Sz, as there are equations, and th% coefficients of the remaining variations must be put separately equal to nothing.* 8. We may suppose the variations Sx, Sy, Sz, in the equation (fj equal to the differentials dz, dy, dz, since these differentials are necessarily subjected to the conditions of the motion of the point M. By making this supposition, and then integrating the equation (J"), we shall havet dx» + dy^dz^_ ^^ g_ f<^P,dx+Q.dy+Rdz, ■ dt* sented by ^a^+b^+c^. If ^-a. f{t)-^hf (t), y=c. f(t)+d.(/'t), z=,.f(t)+g. f (0) the point will move in a curved line; however, this curve is of single curvature; for by eliminating t we obtain an equation of the form a'x-}-6'y + c'2=0, which is the equa- tion of a plane. The simplest case of this form is x=a {t)-\-b {f), y=h (t) -\-d {t^), 2=£. (t) -{-<r (t^), eliminating t between the two first equations we shall obtain an equation of the second order between a; and y, and from the relation which exists between the co- efficients of the three first terms of this equation, it is evident that the curve is a parabola. l[ x=f(t), u^:F{t), ~=:J'J'(t), all the points of the ciu-ve will not exist in the same plane. * The law of the force being given, the investigation of the curve which this force makes the body describe, is much more difficult than the reverse problem of determining the velocity, and force the nature of the curve described being given ; as the integrations which are required in the first case, are much more difficult than the diiferentials which determine the velocity and force in the second case. f We have seen in No. 7, that when a point moves in a right line, its velocity is equa to the element of the space divided by the element of the time ; this is also true when the motion is curvilinear ; for if P.Q.R, the forces soliciting the point parallel to the tliree co- ordinates, should suddenly cease, then the velocity in the direction of each of the coor- dinates will be uniform, and equal to -^> -i^> -^ , respectively, (see second note to at at (it the preceding number) consequently the motion of the point will become uniform, and its direction rectilinear, •.• if v express this velocity we will have, by first note to No. 6. t PART I.— BOOK I. S3 c being a constant quantity. — ^ T-_!: is the square of the ve- locityofi/, which velocity we will denote byu; therefore if Pdx,+ Q.dij, + i2(/2, is an exact differential of a function ?>, we shall have This case obtains when the forces which solicit the point M are func- tions of the distances of their origins from this point. In fact, if ^, 5', &c.* represent these forces, s, s', being the distances of the point M F (See Lacroix Traite Elementaire, No 139.) The rectilinear direction is that of the tan- gent, for if A, B, C, denote the angles which this direction makes respectively with x,y, z, we shall have v. cos. A = -j-, v. cos. B = —^ , v. cos. C = —^ , by substituting dx for V. its value, which has been given above, and tlien dividing we obtain cos. A = — j ds COS.B =-j— , COS. C = — ^; but these are the cosines of the angles which the tangent makes with the coordinates ■.• the tangent coincides with the hne along which the point moves when the forces cease. * If P.dx-\-Q.dy + Rdz =f[x, y, z, ) then u^ =c+2,/ ( j, y, z,) let A be the velocity corresponding to the coordinates a,h,c; then A = c+2. y (a, b, c,) •.• v- — A':m'2. f {x, y, z) — 2./(n, h, c,) •.• the difference of the squares of the velocities depends only on the coordinates of the extreme points of the line described ; consequently when the point describes a curve, the pressure of the moving point on the curve does not affect the velocity. The constant quantity c depends on the values of v, and of x, y, z, at any given instant. When the moving point describes a curve returning into itself, the velocity is always the same at the same point. If the velocities of two points, of which one describes a curve, while the other de- scribes a right line, are equal at equal distances from the centre of force in any one case, they will be equal at all other equal distances. If the force varies as the ^i* power of the distance from the centre, then s and / be- ing any two distances, (p or f(x, y, z,) := /' + ^' •.• v*—A\ s" '^^ — s'"'^^ . In tills case also the differential of the velocity r= s.^ds, therefore by erecting ordinates 34 CELESTIAL MECHANICS, from their origins ; the resultant of all these forces multiplied by the variation of its direction will, by No. 2, be equal to X.SJs ; it is also equal to PJ-v + QJ/j + RJz; therefore we have Pj:i:+QJij + E.Sz=^l.S.Ss. and as the second member of this equation is an exact variation, the first will be so likewise. From the equation (g)* it follows, 1st, that if the point M is not proportional to s", we can exliibit the figure which represents the square o^ the velocity, \rhen n is positive the figure is of the parabolic species, when negative it is hyperbolic. It' the distaiices increase in arithmetical progi'ession, while the lorce decreases in geo- metric progression, the figure representing the square of tlie velocity will be the logarith- mic curve. See Principia Matthematica, lib. 1, prop. 40, 39. If P(/j;-+-Q.(/y-l-iJcfe be an exact differential, then -; — ZZ — — ; — ;— — — ;— ■+■ &c. "^ dj/ dx dz dx P,Q,R, must be functions of ^, y, and z, independant of the time •.• if the centres to which the forces were directed had a motion in space, the time would be involved, and conse- quently P.f/j4- Q.r/y-f/i.f/;:, would not he an exact differential, for then the equations dP dR ^ — 1- &c. would not obtam. dz dx When the forces P,Q,R, arise from friction or the resistance of a fluid, the equation P.dx-\- Q,.dy\-R.dz, does not satisfy the'preceding conditions of integrability, for since P.Q,R, de- pend on the velocities > -j^, — in tliis case ; it is evident that P.dx-\- Q.dt/-\-Rdz cannot be an exact differential of a function of x, y, and ;::, considered as independant varia- bles ••• to integrate P.dx+Q.dy-[-R.dz, we should substitute the values of these va- riables and their difTerentials in a function of the time, which supposes that we have solved the problem ; consequently when the centre to which the force is directed is in mo- tion, and when the force arises from friction or resistance, the velocity is not independant of the curve described. * The velocity is constant when/ (x,i/,z) is constant ; and also when f{x,y,z,) vanishes; when the point is put in motion by an initial impulse, the motion is unifonn, and its direction rectilinear, a.ndv'^ — A'^, = c, — •- = c. -^ =.c", for then -I — — > = P, (it dt dt \ I'f ) {d'lJ 1 I d'z ~> —j-~- ( ^^ Q-'\ ■ I 2 ( = ^^ are equal respectively to nothing. The velocity lost by a body, in its passage from one plane to another, is proportional to PART I.— BOOK I. - 35 solicited by any forces, its velocity is constant, because then iprzO. It is easy to be assured of this otherwise, by observing, that a body moving on a surface or on a curved line, looses, at each rencounter with the indefinitely small [)lane of the surface, or indefinitely small side of the curve, but an indefinitely small part of its velocity of the second order. 2dly. That the point M, in passing from a given point with a given velocity, will have, when it attains another point, the same velo- city, whatever may be the curve which it shall have described. But if the point is not constrained to move on a determined curve, then the curve described possesses a singular property, to which we have been led by metaphysical considerations, and which is, in fact, but a remarkable consequence of the preceding differential equations. It con- sists in this, that the integral. /t'.r/5 comprised between the two extreme points of the curve described, is less than on any other curve if the point is free, or than on any other curve subjected to the same surface if the point is not entirely free. To make this appear we shall observe, that P.dx-^Q.dy + Rdzhemg supposed an exact differential, the equation (^'•) gives f.J'u = P.Sx-irQJy+RSz. in like manner the equation {f) of the preceding number becomes, dx dii d" 0 = §x.d.-^ + Si/.d.-^ + Sz.d. — —v.dt. Sv. dt ^ dt dt naming ds the element of the curve described by the moving point, we shall have v.d(=ds ; ds = .ykt'^+dj/^-tdz*, f2 the tliffercnce between radius and cosine of the indination of the planes, i. e. to the versed sine, or to tlie square of the sine ; and when th.e curvature is continuous the sine is an indefinitely small quantity of first the order, •.• the velocity lost, is an indefinitely small quantity of the second order. 36 CELESTIAL MECHANICS, consequently 0 =li:d.~- + Sy.d. JL + Sz.d.-^ ds.Sv, (li) dt dt dt ' V. / by differentiating with respect to <5', the expression for ds, we have ds . r dx . J , dy . ^ dz ^ r dt dt ' dt ^ dt. The characteristics d and S being independant, it is indifferent which precedes the other ; therefore the preceding equation may be made to assume the following form : , , , (dxSx + dy.Sy + dz.Sz) , ^ f/^r ^ r dy , r dz v.S.ds=d. ^ -^ -^ — ^x.d. — Si/.d. -^ —Sz.d. -— , , dt dt ^ dt dt by substracting from the first member of this equation the second member of the equation (//) we shall have . . , , d. (dxJj: + dy.hi + dz.SzY S irds) zr ^ This last equation integrated with respect to the characteristic d, gives I. fv.ds = const.+ ^^•^^■+^.^%+^^-~-^^ ^ *Ford.i±if±±M±if)^ <I.J^Ss.+d. p^+d^h; +'^d. ^.+ ^ 'l.hj dt dl dt -^ ^ dt at dt + Jld.^x i^j!lhdx+-^hdi,+^^.dz.'l, :■ by performing the operations dt \ dt ' dt ^ dt 3 » > jx , <: dx.lx-X-dy.h/-i-dz-^=\ prescribed m the text, v.e obtain v.d.ds-j-ds.iv=i.{v,dsi= d. i ^ < • This equation being integrated with respect to the characteristic d gives/. d.{v.ds.) const.-)- '^^•'■'+('?Ay+dz-^z _ ^^j^^^^ ^j^^ ^^^.^ extreme points of the curve are fixed, the variations 3x, Jy, 2z, ot the coordinates must be equal to nothing at these points ; con- sequently the variation o{/.(v.ds) is equal to nothing, and •.• r(v.ds) is either a maxi- muni or niininium ; but it is evident from the nature of function /. (v.ds.) that it does not admit a maximum. PART I.— BOOK I. 37 If we extend this integral to the entire curve described by the moving point, and if we suppose the extreme points of this curve invariable, we will have S.J'v.ds = 0, that is to say, of all the curves, which a point solicited by the forces P, Q, R, can describe in its passage from one given point to another, it describes that in which the variation of the integral yt'.cf*, is equal to nothing, and in which, consequently, this integral is a minimum. If the point moves on a given surface without being solicited by any force, its velocity is constant, and the integral fv.ds becomes v.fds. Therefore in this case the curve described by the moving point is the shortest which it is possible to trace on the surface from the point of departure to that of arrival.*. 9. Let us determine the pressure of a point moving on a curved surface. Instead of eliminating from the equation [J") of No. 7> one of the variations Sx, Sy, Sz, by means of the equation to the surface, we can by No. 3 add to this equation, the differential equation of the sur- * Wlien the velocity is constant the integral fv.ds, becomes v. f. ds=v.s ; and since s is a minimum, the time of describing s, which is proportional to s in consequence of the iinifomiify of the motion, will be a minimum in like manner. Since the equation l.J.{v.ds.) =0, has place when Pdx-\-Qfly-\-R.dz is an exact differential, it belongs to all curves that are described by the actions of forces directed to Jixed centres, the forces being functions of tlie distance fiom those centres ; and if the fomi of these functions was given we could determine the species of the curye described, by substituting for v its value in terms of the force, (which we have by a preceding note), and then investigating by the calculus of variations, the relation existing between the coordinates of the curve which answers to the minimum of the expressiony(Ti.rf«). If S the force varied as — ,- by making use of Polar coordinates we would arrive at the polar equation of a conic section, in which the origin of the coordinates would be at the focus of the section ; if S was proportional to s the resulting equation would be also that of a conic section, the origin of the coordinates being at the centre of the section. From the preceding property the known laws of refi-action and reflection have been deduced. Mr. Laplace has also suc- cessfully applied it to the investigation of the law of double refraction of Iceland chrystal, which was first announced by Huyghens, and afterwards confTrmed by the celebrated ex- periments of Malus on the polarization of light. See a paper of Laplace's in the volume of the Institute for the year 1809. 38 CELESTIAL MECHANICS, face multiplied by the indeterminate — xdt, and then consider the three variations Sx, Sjj, Sz, as independant quantities. Therefore let ii = 0 be the equation of the surface, by adding to the equation (J') the term —aSu, (It. the pressure will, by No. 3, be equal to / \du } I dx ) « C f/M / « ) dii )^ I At first let us suppose that the point is not solicited by any force ; its velocity » will be constant, and since v.dt=ds; the element of the time being supposed constant, the element ds of the curve will be so like- wise, and by adding to the equation (./) the term — xJu.dt, we will obtain the three followino- : 0 = V. d'^x { du } ^ , d~y S ^" ds» from which we may collect , d"y \ du I du I ds"- ( dx ) ds^ <^ dy n » d'Z 0 = w. ax but ds beino- constant, the radius of curvature of the curve described by the moving point is equal to ds' t • By substituting for iW its value -r-^^ we eliminate the time i, if the resultin-r equations be squared, we obtain, by adding their corresponding members, _ _. ''■ ^^ j ^ \ dy S "^ \ dz f ■ f This expression for the radius of the osculating curve may be thus investigated : let a, b, c, express the coordinates of the centre of this circle, its radius being equal to r, PART I.— BOOK I. 39 ••• by naming this radius r we shall have * c <r/^' 3 ^ dy ' ^ dz ■> r then r*={x—aY-\-{ij—bY-^[z—cY ; dx. {x—a)-{-dy.{y—b)-\-dz. (2— c), the differential of tills equation is equal to nothing, as any one of these coordiiuites may be considered as a function of the two remaining, we can obtain the following equations of partial dif- ferences dx. {x—a)Jr^z. (z— (•)=0, cii/.{j/—b)+dz. (z—c) =0, (the values of dz in these equations are evidently different,) consequently we have cf'x. (j — a)-{-d^z. (z — c)+dx* +dz^-=0;d-y {y—b)J^d--z.{z-c)-\-dy-^ + dz'==0, V (x-«)=— £ {z—c), (y-b) = dz . . -7—, (z — e), and since ds is supposed to be constant, we have d^x,dx-\-d''y.dy+dfz.dz =0, (d'z in this equation refers to the entire variation of rfz,) consequently z being consi- dered as a function of x and y, we obtain d^x.dx+d^z.dz==0;d^y.dy+d^z.dz = 0; ■.■^=-p^; ^ — ^^ '^ these values being substituted in place of -7^ -; in the preceding equations we shall dx dy have d"^ X d^y •.• by adding together the two preceding differential equations of the second order, sub- stituting for (x — a) (y — b) their values, and observing that the whole variation of z is equal to the sum of the partial ones in these equations, we obtain, ~ . (z — c)+dx- -i-dy- -\-dz'=0, consequently - (dx-+du^+dz'')^ d^ x^ d*y~ by substituting for (.r — aY {y—b^ their values -—-;-. )s — c)- ;■ J^^ .(z — c = ), whicli have been given, we obtain {x-a)-+{y-by-M—cy=.^^^fli'^^T, . {d^x^'-^d^y^^H') d^x^-\-d'yi-d- 1 ds'' 4,0 CELESTIAL MECHANICS, consequently the pressure which the point exercises against the surface is equal to the square of the velocity divided by the radius of curvature of the curve described. If the point moves on a spheric sui'face,* it will describe the circum- ference of a great circle of the sphere,, which passes through the pri- mitive direction of its motion ; since there is no reason why it should deviate to the right rather than to the left of the plane of this circle ; therefore its pressure against the surface, or what amounts to the same, against the circumference which it describes, is equal to the square of the velocity divided by the radius of this circle. If we conceive the point attached to the extremity of a thread desti- tute of mass, having the other extremity fiistened to the centre of the surface, it is evident that the force with which the point presses the circumference is equal to the force with which the String would be tended if the point was retained by it alone. The effort' which this point would make to tend the string, and to increase its distance from the centre of the circle, is denominated the centrifugal force ; there- fore the centrifugal is equal to the square of the velocity divided by the radius. The centrifugal force! of a point moving on any curve whatever is * If the point move on a spherical surface, the motion will be necessarily performed on a great circle, for the deflection can only take place in the direction of radius, and in the plane in wliich the~body moves. -f- If the body moves on any curve whatever, the centrifugal force =: — , this force acts in the direction of a normal to the curve, and if all the acceleratiag forces which act on the point be resolved into two, of which one is in the direction of the normal, and the other in the direction of the tangent, the resultant of the centrifugal force, and of the former of these decomposed forces, is the entire pressure with which the point acts against the curve, and the resistance of the cui^ve is an accelerating force equal and con- trary to this resultant. If we denote this normal force by L, and if A, B, C, be the angles which it makes with the coordinates x, y, z, respectively, then by the equation (y) and No. 3, we have '^=P+L. cos. A; -^ = Q+L. cos. B; ~ = R+L. cos. C; - PART I.— BOOK I. 41 equal to the square of the velocity divided by the radius of curvature of the curve ; because the indefinitely small arc of this curve is confounded with the circumference of the osculating circle. Therefore we shall and since -J—, —--' —j- , express tlie cosines of the angles which the tangent makes" with X, y, and s, ^-. cos .<4+-,- . cos.i?+— '^. cos. C.=0; because the tangent is per- ds - lis as ' pendicular to the normal. (See last note to No. 1). We liave also cos ^A-\- cos. -B-\- cos. ^C=l, and the four undetermined quantities L, A, B, C, being eliminated between the five preceding equations, the resulting equation will be one of the second order be- tween x, y, I, and / ; this equation combined with the two equations of the trajector}' which are given in each particular case, are sufficient to determine the coordinates in a function of the time. See notes to No. 3, and No. 7. The elimination of L, A, B, C, might be effected by one operation ; for multiplying the three preceding equations by dx, dy, dz, respectively, and adding them together, we obtain the following equation : ^' dt^ ~' '^' ~ ^•'^^+ ^•^'/•+ ^'^--i- ^- ('^os- ■^•(1^+ COS. B.dy+ cos.C.ffe.) (the latter part of this second member is equal to nothing, as has been already remarked ;) and since ds-=dx~~ -\r d ij\ ^ dz'- , d''s.ds=d''x.dx-{-d^y.dy+d^z.dz, ;• we shall have d's _„ dx dy d^ df^-^-ir^^-'dT^^-ds ' from this last equation it appears that tlie accelerating force resolved in the direction of the tangent, is equal to the second differential coefficient of the arc considered as a func- tion of the time, •.• this expression for the force has place whatever be the nature of the line along which the point moves. See Notes to No. 7. In like manner it appears that the expression for the force in the dii'ection of the tangent is altogether independant of L. d's It is also evident, that when there is no accelerating force -j-j- = 0, this also follows from the circumstance of the velocity being uniform when P, Q, R, are equal to notliing. Let V denote the resultant of all the accelerating forces which act on the point, and 6 the angle which this resultant makes with the normal, then V. cos. 6 will be the ex- pression of the resultant resolved in the direction of the normal ; and when all the points of the curve exist in the same plane, the entire pressure will be equal to the sum or dif- ference of -- — , and V. cos. 6, according as these two forces act in the same or in con- 42 CELESTIAL MECHANICS, have the pressure of the point on the curve which it describes by add- ing to the square of the velocity, divided by the radius of curvature, the pressure produced by the forces which solicit this point. *t traiy directions, •■• +/,=: =fc: \. V. cos. «. We can express this pressure otherwise by means of the rectangular coordinates ; for since P, Q, are the expressions for the force V resolved parallel to x and _?/, these forces resolved in the direction of the normal are equal respectively to P. -j-; Q. —z—, (the signs of — -, and -j-, are evidently dif- ferent) consequently we have r. cos. 6=JfP.-f- + Q.— , and v L = — -h P. -f- -fQ. -j-, as (Is r as as therefore if we know the equation of the trajectory, and if we have also the values of P and Q in terms of tlie cooi-dinates, we can determine the velocity, and consequently L, and d'x d^v d^z substituting this value of L in the expressions for -j— ' , ' — ~ , which liave been given in the foregoing part of this note, we might b)' integrating determine the velocity in the direction of each of the coordinates, and also the position of the point at a given moment. If the point be attached to one extremity of a thread supposed without mass, of wliich the other extremity is fixed in the evoluf e of the curve described, then the point receiving such an impulse, that the string remaining always tended, may unroll itself in the plane of the evo- lute, it will describe the given curve ; the direction of the string is always perpendicular to the curve, and its tension is equal to the normal pressure on the trajectorj^, and conse- 1 ■"* . P^du + Qdv „ ... . - ^ ^ ,. quently equal to 1 ^ . By equating this expression ot L to notmng, we can derive the equation of those trajectories in wliich the motion is fi-ee, or in which the trajectory may be described freely, i. e. it is not necessary to retain the point on the curve by means of a ' thread, or a canal, or any perpendicular force. * If the motion is performed in a resisting medium, this resistance may be considered as a force acting in a direction contrary to that of the motion of the body, consequently it must tend to some point in the tangent. If we denote tliis resistance by / its moment is equal — 7.Jj (j = >/{.T- 0' + (.y — "')'+C~ — ")*' ^' "'j "> ^6 the .coordinates of the cen- tre of the force 7. therefore 3J = ^-^^. Sx-f-^i!=^^. ^y+ ^^^^. h; if we suppose .r — I dx the centre of force in the tangent, then i— \/dx'+di/^+ds'' =ds •: — ~ — "^ ' •*,. PART I— BOOK I. 43 y— m^_ J) _ z—i_ __ _ ^^j j^. _ j^_^ — 1^^ J _^ — ^ ^ j^^ .^^j^g resisting nie- i ds t as as as as dium was in motion, its motion must be compounded with the motion of the body, in order to have the direction of the resisting force. If da, db, dc, be the spaces described by the medium, wliile the body describes ds, these quantities must be added or subducted from dx, dy, dz, in order to have the relative motions, and as ds = y/dx'' +dij*-{-dz^, if Me dx — da . make d(r ;^ -/{dx — da)'--i-{cli/ — db)'-]^(dx — dc)*, we shall have Si = — '^ °'^+ ^ . jwJ — 1 —, ^z. Tlie resistance / in general= i|/ (v), a function of the ve- dr ■^ da- locity, in this case it is a function of the relative velcoity. By the preceding investigation we ai-e enabled to apply our general formula to motions made in resisting mediums without entering into a particular consideration of this species of motion. However the analysis becomes very complicated when the forces which com- pose P, Q. R, exist in different planes, and as in this case, the causes on which the va- riation of the velocity depends, arise in some measiu-e from the velocities themselves, we are not permitted to regard P.dx+ Q.d^ i-R.dz, as an exact differential of three inde- pendant variables, which facilitates our investigations when the motion is performed in a vacuo. See Notes to Nos. 8. We might also reduce to our general formula, the differential equations of motion, when the retardation arises from the friction against the sides of the canal. f If the body moved on a surface we might, as before, abstract from the consideration of the surface, and consider the material point entirely free by adding to the given forces anotlier accelerating force, of lohich the intensity is unknown, and of which the direction is normal to the surface, •.• if this force be denoted by L we shall have, by the equation (y ) of No. 7, and by No. 3, the following equations : (m^O is the equation of the suface. See Notes to No. 3). If we eliminate L between these three equations, N will also disappear ; and if the two differential equations of the second order, which result from this elimination, be combined v/ith the equation tc=:0 of the surface^ we can detennine the tliree coordinates of the point in a function of the time. If we multiply the preceding equations by dx, dy, dz. respectively, and then add together the corresponding members, we will obtain d^x.dx+d'^M.du+d'z.dz „ , ^ , „ , ,t ^ f ^« 7 , . f ^" 1 > ■ -~^^ = P.dx+Q.dy+Rdz+N.L. \ f^ \ '^'■'+ \-^ \ '^V^ f 3u 1 \ Y7 W« ; but the last part of the second member is = to nothmg, ' g2 44 CELESTIAL MECHANICS, When the point moves on a surface,* the pressure due to the centri- fugal force, is • qual to the square of the velocity, divided by the radius of the oscuiati .g circle, and multiplied by the sine of the inclination of the plane of this circle, to the plane which touches the surface ; therefore, if ■ ve add to this pressure, that which arises from the action of the forces which solicit the point, we shall hiive the entire force with which the point presses the surface. |sincerfi< = 0, and -! -^ | = i -1 V v if V.dx-\-Q_.dy+R.dz is an exact difFeren- tial, we shall have —r;= P-—, — h Q.~- -\-R.^, as before, and— — = r- = C4- dt^ as dt ds dt^ y{P.dx-\-Q.d^-f-R.dz), audif P,Q,R, and consequently v were given in tenns of the coordinates, we might obtain immediately the differential equations of the trajectory by d^x f ^71 i ' multipljing the equation ---^;-= P+L.A^. ■} -r — j- , by di/ and dz successively, and cit~ (. dx ) then subducting it from the two remaining equations multiplied by dx ; by concinnating ds the resulting equations, substituting for dt its value — , and for i; its value m a function of the coordinates, we obtain two differential equations of the second order, fi-om which eliminating the quantities LN there results a differential equation of the second order be- tween the three coordinates z,y,z, solely; this equation, and the equation «^0 of the surface will be the two equations of the trajectory. * If a point moves on any curve the centrifugal force is always directed along the radius of the osculating circle ; and since the pressure on the surface is always estimated in the direction of a normal to the surface, (see No. 3) if the plane of the trajectory is not at right angles to the surface, the radius of the osculating circle will not coincide with the normal to the surface, and consequently the part of the centrifugal which pro- duces a pressure on the surface is equal to , multiplied into the cosine of the an- gle which the radius makes with the nonnal, but this angle is evidently the comple- ment of the angle which the plane of tlie osculating circle makes with the plane which touches the surface. If the forces soliciting the point are resolved into two, of wliich one is perpendicular to the trajectory, then the resultant of this last force, and of the ccntrifijgal force, will express the whole force of pressure on the curve; if this curve was fixed, it would be sufficient for the pressure to be counteracted, that its direction was in a plane perpendicular to this curve , but if the curve be one traced on a given surface, then, in order that the pressure should be counteracted, it is necessary that the resultant of the forces should be in the direction of a nornml to the surface. Sec note to page 16. PART I.— BOOK I. 45 We have seen that when the point is not solicited by any forces, its pressure against the surface, is equal to the square of the velocity, di- vided by the radius of the osculating circle ; therefore the plane of this circle, that is to say, the plane which passes through two consecutive sides of the curve described by the point is then perpendicular to the surface. This Curve on the surface of the earth is called the perpen- dicular to the meridian ; and it has been proved (in No. 8) that it is the shortest which can be drawn from one point to another on the surface.* * If we make the axis of one of the coordinates to coincide with the normal to the surface, we can immediately determine the inclination of the plane of the osculating circle to the plane touching the surface ; for if we denote by A, B, the angles which the radius of the osculating circle makes with the normal and witli the coordinate which is in the plane of the tangent, and by »i, n, I, the angles which the resultant V of all the forces makes with the three coordinates, the force _ll resolved parallel to these coordinates is equal to - — . cos. A, —. cos. B, A '-. COS. 100°, (because the angle between the radius and r r r tangent to the curve is equal to 100') in like manner the force V. resolved parallel to these coordinates equals V. cos. m, V. cos. n, V. cos. I, since A and m denote the inclination* of the radius of curvature, and of the resultant to the normal, . cos. A-\-V. cos. m, r express the pressure of the point on the siuface, V. cos. n-\-- — cos. 100°, or V. cos. n is the force by which the body is moved ; and since this motion is performed in the di- rection of the tangent, V. cos. l-\ . cos. B, which expresses the motion perpendicular to the tangent must vanish; consequently we have V. cos. /-|- cos. B~0, '.' if F. I, V, and r were given we might determine B, which is = to the inclination of the plane of the osculating circle to the plane touching the surface, it also foUoTvs, that when the point is not soUcited by any accelerating force, cos. B=0, •.• B= 100", or the plane of the osculating circle is perpendicular to the surface, which we have previously established from other considerations. Ifthe plane whose intersection with the surface produces the given curve is not ■perpen- dicular to the surface, then the radius of curvature is equal to the sine of the inclination of the cutting plane to the plane touching the surface, multiplied into the radius of cur- vature of the section made by a plane passing through the normal to the surface, and tlirough the intersection of the plane touching the surface and the cutting plane. See LacroLx, No. SSi. ••• the pressure is the same whether the point move in a greater or 46 CELESTIAL MECHANICS, 10. Of all the forces that we observe on the earth, the most re- markable is gravity ; it penetrates the most inward parts of bodies, and would make them all fall with equal velocities, if the resistance of the air was removed. Gravity is very neai'ly the same at the greatest heights to which we are able to ascend, and at the lowest depths to which we can descend ; its direction is perpendicular to the horizon, but on ac- count of the small extent of the curves which projectiles describe rela- tively to the circumference of the earth, we may, without sensible error, suppose that it is constant, and that it acts in parallel lines. These bodies being moved in a resisting fluid, we shall call b the resist- ance which they experience ; it is directed along the side ds of the curve which they describe ; moreover we will denote the gravity by g. This being premised, let us resume the equation (fj of No. 7, and suppose that tke plane of x and y is horizontal, and that the origin of ^ is at the most elevated point ; the force b will produce in the^^direction of the coordinates .r, y, z, the three forces — b. — , — b.-~ , — b.-^ •.' by ds ds ds No. 7 we shall have F=z—b. ~ ; Q--^b.^ ; R-^b.-^ +g-. * dx ^ , du -o 1 dz ds ds as and the equation CJ]) becomes 0=^;..^^. 't+b.^di.l+Sy.\d.± +b.^dt.l I dt^ ds S ^ ^ dt ^ ds ) ^■Sz.\d.^x.b.~ dt.—g.dt. I * I dt^ ds ^ $ less circle, for the sine of inclination occurs both in the numerator and denominator of the expresiion ; this also follows from considering the proportion of the sagiita of curva- ture in a peipendicular and oblique plane. The investigation of the shortest line which can be drawn between two given points on a curved surface, whose equation is u—0, by the method of variations, leads us to the same conclusions. ?ee Lacroix. The consideration of the shortest line which can be traced cm a spheroidical surface is of great importance in the theory of the figure of the earth. (See Book 3, No. 38.) * Since the force b acts in tlie direction of the tangent or of the element ds of the curve (see note to No. 9,)^ this force rs'solved parallel to the three coordinates jc, y, z, dx dy dz ^ dx di) dz , . ^ , = "•~-ri"-~T'i O'—j— , \ot —r- , -y- , — — are =: to the cosmes of the angles as ds ds ds ds ds " PART I— BOOK I. 4? If the body be entirely free we shall have the three equations , d.v , dx , ^ 1 dy , , dij ,. Q—d,—— +b.---.dt; 0 = d.-~- +b. -^. dt^ dt ds dt ds 0 = d.-^ +b.~.dt—g.dt, dt ds The two first give ±. d.±-^. d. <^ = 0. dt dt dt dt from which we obtain by integrating, dx^=.fdy, f being a constant arbi- trary quantity. This equation belongs* to an horizontal right line, therefore the body moves in a vertical jjlane. By taking this plane for that of x, z, we shall have ?/=:0, the two equations, , dx ^ , dx , ^ , rfs . , dz ,^ J, 0 = d.- hb. -r-.dt; O^d.—-. +b.—-~. dt — g. dt, dt ds dt ds will give, by making dx constant, , ds.d»t ^ d*z dz.d*t , , dz , '=-dF~' ""—dT rfF- +'• ^•^^-^••^^^- From* which we obtain g.dt* = d*z, and by taking the differential which the tangent makes with the three coordinates ; they are affected with negative sign* because they tend to diminish the coordinates. ^ _. ... dt/ , dx dx , dii , tfi/* . , * Dividing -f^. d.— . a.— T- =: 0, by -^ it becomes ^ dt dt dt dt ^ df^ dx ^'\~, ^^^^' ■•■ '^y integrating -r— =y" and dx==f.dy, since tlie equation of the ~dt projection of the line wliich the projectile describes on a horizontal plane, is that of a right line, the body must have moved in a vertical plane, otherwise its projection on an horizontal plane would not be a right line ; this circumstance we might have anticipated from the manner in which the forces act on the body. ^ -wc 1 ', , , dx dx dx,d^t * It we make dt constant in the equation d. ^- + i. -=- . di= 0, we get -; dt ds dt- 48 CELESTIAL MECHANICS, . ^g.dt.dH=d'z, if we substitute for rf»nts value Af[fl, and for dt* hi ds d*z - value . — —, we shall have b ds.d^z i~\% g 2(rf This equation gives the law of the resistance b, which is necessary to make the projectile describe a given curve. If the resistance be * proportional to the square of the velocity, b is ds* equal to h, , h being constant, when the density of the medium is dt* ' , unifoiTO. We shall have then b h.ds* h.ds* g g'dl* d*s. d^z • ■ d*" therefore h.ds= , which gives by integrating — ~ = Sa.c^.t 2.d*z da.* + J.— . dt]= 0, '.• b — — j — , by substituting tliis quantity in place of b, and differen- tiating, we get the expression d^z dzJH ds.d't dz , , d^z dz.d^f , dzJH , -Jt dir +-11^ '■d^'^-s-^' = —t dF- + -d? -"'^'•= d'z -g.dt= 0, •■•by differentiating we obtain d^z ='2g.dt.d^f, and substituting for dt d't its value 4. — ~, and for dt' its value — , we arrive at the following equation, ds g 2g.b. C d'z -\ ^ b dH ds I g i g ^d^r * The value of the consrant coefficient /; is obtained by experiment ; it is different in different fluids, and when bodies of different figures move in the same fluid. ds^ t Since the square of the velocity is equal to -^-j-, the resistance is expressed by ds' , d'-z hds^ ds.d^z d-z h.-j^, vby substituting for rf<= its value — ^ 'WT^ 2{d>z)*'' "'" "" "rf»7 ' PART I.— BOOK I. 49 a being a constant arbitrary quantity, and c being the number whose hyperbolic logarithm is unity. If we suppose the resistance of the me- H ••• 2A.4 = log. d^z^log. F :■ g^"^ ^F.d'z, v -— — = -— . (Let 2a= -^r— ) and we r.ax^ ax- F.d.r:^ shall have 2ac-*' = ^ ; dx being constant it is permitted to introduce dx- as a divisor. The constant quantity a depends on the velocity of piojection, and on the angle which its direction makes with the horizon ; for by substituting — g.dt^ in place of d-z we dx^ n- dx^ shall have ■ — ■= -2 — .c— 2'« is the velocity of the body in the direction of the axis of x at the end of the time t ; let ii be the velocity of projection, and i the angle which its direction makes with the horizon, we shall have at the same time t = 0, .T=:0, 2=0, and —r- = u. cos. 6, :■ ii^. cos. *tf = — -~- . Let h be the height due dt 2a ° to the velocity u, u^ =%/'> *•■ by substituting for u- its value, we deduce a= 4:k cos.^# By making dz=pdx, ds becomes equal to dx.^i-i-p'- , •■• — c-''K ds=2h. cos.* t. dp. c d^z 7 c2*s. VI +i^%i""' ''P = -J^ 5 *•• by integrating — — \-C=2h. cos.^ S.fdp.^l+p\ I =2A. COS.' 6.f-^^J=^ + 2h. cos. e.f J^^ } ='''• COS.' 6. log. (p + ^/H:^), -|-.A. COS.- 6. p. \/l-[-p-, the constant quantity C is easily found; for since p is the derivitive function of z considered as a function of x, at the commencement of the motion, when 4=0, p~ the tangent of the angle of projection wliich is given, '.• C is equal to h. cos.=^ 6. } (log. (tan. <-f-sec. 6) +tan. 6. sec. 6.) i + -y-. By substituting for ; — its value, which we obtain from the equation r— = 2.. 2A 2n <ip . , cos. 2 (. — i-, we deduce ax dx =z , and rfz =r '^/'•[^log.(/;+-^/l+p^)^-/J.^/ 1+/; j_q p.dx=z and since g.dt^=d^z=dp.dx we have <^<»= 2Aillog. (p+v/l+/^')+;;.x/l+p^)-d 50 CELESTIAL MECHANICS, dium to vanish, h is equal to 0 ; then by integrating* we will obtain the equation to the parabola 2;= ajr*+/'^+c, i and c being constant arbi- trary quantities. The differential equation d'zzzg.dt^, will give dt'= dx^, from g v^,= '^P — X {'2gh (log. (/;+^l+;;r)+;;Vl+P^)— <^] "• If the integrals for these values of dx, dy, dt, could be exhibited in a finite form, the pro. blem would be complete!)' solved, for the integrations of tlie two first equations would give the values of x and 2 in funciions of p ; and if p be eliminated between the resulting equations, the relation between x and y would be had ; those integrations have hitherto baffled the skill of the most celebrated analysts. However by means of the expressions for dx and dz, we can describe the curve by a series of points, and the approximation wilj be always more accurate, according as we divide the interval between the extreme values of^ into a greater number of parts. We might collect some of the remarkable proper. ties of the curve described from the preceding values of dx, dz ; for if p be very great, log. dp iP^V^-VP') vanishes with respect to;;, and '.• the limit of dx, dz, and dt are ^ — —, dp ''P 1 , , " 1 1 2^' """'^ >/5^ ■•■ ^^ ™'^S'''^S "^^ S^^ ^'="- p' - =" "^ '''^- ■?'' '^^ + 2P" • log.;?, the first equation indicates that.j; has a limit, the vertical ordinate increases inde- finitely, but in a less ratio than ;;, therefore'the descending branch has a vertical asymp- tote. By eliminating log. ;j in the expression for t we get an expression for z fiom which ■we may collect, that according as the direction of the motion approaches towards the vertical, the motion of the body tends to become uniform. When the angle of projection is very small, we can find by approximation the relation which exists between .t, and 2, for that portion of the trajectory which is situated above the horizontal axis ; in this case the tangent is very neai-ly horizontal, •.• p is very small, and Vl+p =l,y.;;.-.-f/« = r/aVr+7- =dx, q.p. and s=,t, for they commence together, and substituting x in place of ^, we have ~ = — gT^^ ^"^ " ^^'"^ ^^ hypothesis very small, cos.^ (t =-- 1, •.• dp =—^-. dx, by integrating this equation, when we know the Ik value of the constant arbitrary quantity which is introduced by the integitition we obtain the value of p and •.• of 2 =j p.dx. See a memoir of Legendre's in the Transactions of the Academy of Beriin for the yeai- 1782. • In this case-rr =5ia, v -^=2ax-t-i, v s=ax'-|-i.«+c. d.X' dx PART L— BOOK I. 51 which we may obtain t-zix. V ■ — +f'- If *> •s, and t, commence to- gather, we shall have c= 0, f'= O, and consequently g which gives 2 '2a These three equations contain the whole theory of projectiles in a va- cuum ; it follows, from what precedes, that the velocity is uniform in an horizontal direction,* and that in the vertical direction the velocity is the same as if the body fell down the vertical. If the body moves from a state of repose b will vanish, and we shall have dz _ _ 1 therefore the velocity acquired increases as the time, and the space in- creases as the square of the time. It is easy bymeans of these formula to compare the centrifugal force with that of gravity. For v being the velocity of a body moving in the circum- ference of a circle, of which the radius is r, it appears from No. 9, that its ^» centrifugal force is equal to . Let h be the height from which the body must fall to acquire the velocity v ; by what precedes we shall have v' = 2g.h ; from which we obtain — «•. zJ—. The centrifugal dx I fJ^ * ^^^ Hr ~ ^^ velocity in an horizontal direction =\/ - --, and -^ = the velocity V ^ci (It in a vertical direction = gt ,b.\/ _£.. 52 CELESTIAL MECHANICS, force will be equal to the gravity g, if h^ Therefore* a heavy body attached to the extremity of a thread, which is fixed at its other extremity, on an horizontal plane, will tend the string with the same force as if it was suspended vertically ; provided that it moves on this plane, with a velocity equal to that which the body would acquire in falling down a height equal to half the length of the thread. 1 1. Let us considej the motion of a heavy body on a spherical surface, denoting its radius by r, and fixing the origin of the coordinates at its> centre, we shall have r' — r' — j/' — ^'=0; this equation being com- pared with that of z^— 0, gives u = r' — a'" — y'' — z" ; therefore if we add to the equation (X) of No. 7, the function Su multiplied by the indeterrai- nnte quantity — x.dt. we shall have 0=Ss. S'd. — + 'ixx.dt. I + ^i/. I d.-^ + 2x.ij.dt. l + Sz. I d. —^ Q.xz.dt—g.d/. I * In this equation we can put the coefficients of each of the variations Sx, Sy, Sz, equal to nothing, which gives the three following equations : Q^d. — + 2\.xdt, dt 0 = d. ^ + 2x.y.dt. dt + ^' d^ O = d. — ^o- 2k.z.dt — g.dt, dt * The plane of the motion behig horizontal, the force with which the string is tended arises entirely from the centrifugal force. t Po.{^|=-...{^ ;=-.,. m=_.. PART I.~BOOK I. 53 The indeterminate a makes known the force with which the point presses on the surface. This pressure by No. 9 is equal to consequently it is equal to 2xr ; but by No. 8 we have _ dx^+dy'^+dz' c+2gz ^ * dt* c being a constant arbitrary quantity ; by adding this equation to the equations {A) divided by dl, and multiplied respectively by x, y, z, and then observing that x.dx+y.dy-\-z.dz =0, x.d''x+y.d^i/-{-z.d'z + dx^ + di/''+dz9=0, are the first and second diflPerential equations of the surface, we shall obtain* * For performing these operations we get c-{-2gz = d? + "rfF ^d^"^ -5^+2^-(^=+y'+s')-g2, therefore we have SAr'' = c-\-?igz, and '2xr = ^^ , •.• the pressure is equal to "*" ° , when the ini- tial velocity c vanishes, the tension of the pendulum vibrating in a quadrantal arc is, at the lowest point, = to three times the force of gravity ; — = the cosine of the angle wliich the r radius r makes with the vertical, therefore it follows that when a body falls from a state of rest, the pressure on any point is proportional to the cosine, of the distance from the lowest point, it is easy to collect, in like manner, that the accelerating force va- ries as the rigiit sine of the angular distance from the lowest point ; we might from the preceding expression for tlie pressure deternune the point where this pressure is in a given ratio to the force of gravity. 54 CELESTIAL MECHANICS, If we multiply the first of the equations (A) by — t/, and add it to the second, multiplied by x, and then integrate their sum, we shall have dr.rfj/ — V'dx _. * dt "^^ d being a new arbitrary quantity. Thus the motion of the point is reduced to three differential equa- tions of the first order, x.dx-\-y.dy = — z.dz, x.dy — y.dx = c'.rf?, dx* >- di/' + dz* ,a -J—J- = C + 22-Z. By squaring each member of the two first equations,t and then adding them together, we shall have i^^+y^) (dx^ + dy*) = c''dt' + z'dz\ * x.dy — y.dx = c .dt shews that the area described by a body moving on a spherical surface, and projected on the plane x, y, is proportional to the time ; the same area pro- jected on the plane x, 2, or y., z, is not constant in a given time ; for if we add to the first of the equations {A) multiplied by — z, the third multiplied by x, and then integrate x.dz — z,"x y , their sum, it becomes equal to = c' ■\- f.{gx.d(), this might have been anti- cipated, as the force ^ does not pass pei'pelually through the origin of the coordinates, ••• x.dz — z.dx, y.dz — z.dy are not proportional to the time, but as there is no force acting parallel to the horizontal plane, x.dy — y.dx must be proportional to the time, f For we have in this case x'^ .dx'' ■\-y'^ Ay'' .\-2x.y.dx.dy = z'^.dz''. x^dy^+y .dx^ — ^x.y.dx.dy = c'*rfr. \'(x''+y'^){dx'' + dy'') =^~c'''dF+z^Mz^. dx^-{-dy'^ :• by substituting for x'^+y*, and 'J~^ ' ^^"^ values we obtain (r* — «») . PART I— BOOK I. S5 If we substitute in place of.r'+3/% and '^' ' -^ ^ their respective dz* values r* — z\ and c + 2^'-2r ~rr''> "^^ ^'^^ '^^^^ on the supposition that the body departs from the vertical dt — ^(r'—z'). (c + 2gz)—c'\ The function* under the radical may be made to assume the form (a — z). (b — s).(2o's+y) ; a, b,f, being determined by the equations {cdt'+2gz.dt^—dz^) = c'^dt-+z\dz^, therefore (r«— =*) . {c+2gz) — c ^). dt^ = r'.rfz^ +Z-&* — z'^.dz'^, consequently — r.dz dt = ^/{r^—z').(c+2gz)—c'^, dz is affected with a negative sign, because tlie motion commencing when the body Is at the lowest point, ^ decreases according as t increases. * If we multiply the factors of the expression, and range them according to the dimensions of z, .we get — 2gz^ — c2^+2r^.g2+/-'c — c'S if the same operation be performed on the expression (a— z).(z — b) . (,2gz+f) we will obtain — 2gz^+ {2g (a + b) — /). 2- + (y. (« + 6) — 2g.ab) z—^fab, these two expressions being always equal, their corresponding terms must be identical, consequently, by comparing the coefficients of z, we have y = 2g. - — -- — - by comparing the coefficients of s', and substituting for f its value we get 2.g -( (a+A) --r— J- = — <^ '•■ ^y concmnaung a-'-\-2ab-lfb--—r''4-ab „ f r^—a^ah—b^ 1 2a. 1 — ! — — C = 2g. < — — ; f the comparison of the absolute quantities, gives, by substituting for / and c their values, which ha*e been already found, —r'-.b^-+>-^.ab4-a''b- ) „ (r'—a').{r- 22. < ' ' J- = Jg. = ' <:; 56 CELESTIAL MECHANICS, a+b We can thus substitute for the arbitrary quantities c and d, a and b, which are also arbitrary, of which the first is the greatest value of z, and the second the least. Then, by making • « /^=^ sin. 9= V r ' « — Z> the preceding differential equation will become dt= r.^lTiTV) d^ these values of^ c, c' being possible, we are permitted to substitute the expression {a~z) .{z—b) .{<2.gz-\.f) in place of (r^— 2;-) . (c+2o;r)— e'S therefore ^ = — v/(^a — 2^ . [z—b) . (^ ^^; ^rJ)> ^ being a function of/, this differential coefficient vanishes when a=s, and also when z=b; — = — ^ (^a — z).{z — u) K^g^-rJ) — 0, has at least txao real roots ; for as the point is constrained to move on the surface of the sphere, the trajectorj' has necessarily a maximum and a minimum ; and as impossible roots enter equations by pairs, it follows that all the roots are real, moreover it is manifest from the variations of the signs, tliat one root is negative: _lf expresses the velocity of the point in the direction of the vertical. * The transformation sin . « = a/ is made in order to facilitate the integration. ^ a — b sin.* i = °~^-, and cos.* 6= ^^^^ v z=:a. cos. ' «+6.(l— cos.* ^) = « cos.^ d-\-b sin.^ t, a — 6 a — b PART I.— BOOK I. 57 y* being equal to (a-\-b)i+r* — b', ~ The angle 6 gives the coordinate z by means of the equation ; z=a. COS.* 9+ i. sin.* 6, dt. COS. «r: ■ == •% —dz=: 2dl. ^/ (a~z) . (z—6) —r.dz and ' - ^^^ ^ (substituting for /its value) -/ (a-zXz-b).{2gz+f) ^ J . ] 2rJe;^(a—z).(z—f>) 2r.d« (substituting for e its value a. cos.M+5. sin.= «) we obtain 2r.de \/a+b \/2g.(a".cos. ^S-j-ab.cos.'6^ab.sin.^e+b~.sin.*i+r*'i-ab 2r.de.^/a + b 2r.d6.\/7+b v/2g.(a-t-6)'-)-l?--— 6^)+(A^— a'). sin.»«)' for 4*— a - in the preceding expression we shall have dt = ""*''*" ^'=(7HiH^»^=)' ^'-'''=-((«+^)'-+('-«-6')).yS .-. substituting 2r.di.^ya+b v/%U«+<' J-i-C-'— 6--)— ((a-t-6)^4-(r-~6^)).y».sin.M. _ r.v/2.(»+i) di 58 CELESTIAL MECHANICS, and the coordinate z divided by r, expresses the cosine of the angle which the radius r makes with the vertical. Let TB- be the angle which the vertical plane passing through the radius r, makes with the vertical plane which passes through the axis of J j w« shall have XZZs/l^ 2*. COS. -ar; * y — oj T^—H^SWi. w; which give xdy — ydx=.{r'^'—!?'). d-cr, .'. the equation xdy — ydx—c'dt will give d.dt dTszz. r'—z^ we will obtain the angle Ts-in a function of S, by substituting for z and dt their preceding values in terms of 6 ; thus we may know at any time whatever, the two angles 9 and -cr, which is sufficient to determine the position of the moving point. T Let us name, — , the time employed t in passing from the greatest * X = the product of the projection of r, on the plane x, y, into the cosine of the angle which x makes with the projected line, .•. as, \f r' — z'- =rr so projected, and » = the angle which xmakes with ^Z r'- — 2^, x=icos. w. \/r» — i^, rfx= —\'r'^ — s'. sin. w. zf/z. cos. OT / — . , /-:; 1 , *dz. sin. v» d^, — . , y =■ \f r'^ — z'.sin. w, .•. dy=\/r^ — s^.ttm. cos. o — 2 . , , xrfz. sin. «7. cos. 33- , , , ., , xdy—ydi=\j^—z^) d'a cos. »ot. h (r«— 2-)rf«. wa z.dz. sin. ra cos. «r. , , , ^ j f For evolving the expression for dt into a series, it becomes, i.V«.sin.'«.rf«+|^V*.sin.*.«.... . 2^g '••^^'"+^'' t/<i+ i.v«.sin.'«.rf«+i^v*.sin.«.«.rf«+^v«. Bin.* «*+*c. v/^- ((,a + 6)*+r»-6» COS. 2« 1 . cos. 4 « 4. COS. 2« but sin. •<= ^ + -2, sin. ♦#=— g 5— + 2;^ PART I.— BOOK I. 59 to the least value of z, a semi-oscillation. In order to determine it, we should integrate the preceding value of dt from 9=0 to flzij.Tr, w I 2 . . COS. 6* , 6. COS. 4« 15. COS. 2« , 10 '""•'= 32-+— 32 32— +3-2'*'=- 4. -r 2'^ • — 32 16 ^2.4' ^. . , sin 6* , 6. sin. ■i^ 15. sin. 2«, 10 « „ /^^-•'•'^*=— r92-+-i28 6r-+ 12-' '"'- (See Lacroix, Traite Elementaire, No. 200.) These quantities being integrated between the Umits fl^O, and 6=^. n, or between sin. 1=0, and sin. fl=l, i. e. between the greatest and least values of z, become respectively «r 1 3x1 lOx 1 r3.5.!r l'?^^, .,-.v- —.—,r-—i, -——•—= \ ~ .— , S- &c. for the parts m wluch the Sines of the multiple arcs 2 I 2.*.2 32 2 (_ 2.4.6 23 occur, vanish, being respectively = to sin. (2ir), sin. (ix), sin. (6ir), the numeral co- efficients of ~ are equal to the corresponding coefficients in the expanded radical ; .*. these integrals being substituted in the preceding series we obtain 2 Vg. V (a+6J^+lr»_6^) [2 "^ 2'^ 2-2 +2.4"^ "teg 5 , 1.3.5 , /1.3.5 :r. , ^ + 2:1:6^ We 2)+^''- Ifin the series, dtJf.-^ . y». sin. 'e.di+~ y*.sin.*«. dS+-^-^.y'^ sm.^i.d64-&c. theintegrali * ^.* i2.4.D being taken as above, between the limits sin. <—0,sin.tf=:± 1; «^ii, «=-— (271+1). »•, (where A- and n are any numbers whatever) will satisfy these conditions ; from which indeter- mination of k and n, it follows, that the vertical coordinate passes through its maximum and minimum an indefinite number of times, and consequently, when all obstacles are removed, the number of oscillations is infinite ; we would obtain an expression for the time intervening between tlie commencement of the motion, and the successive transits through the greatest 13 5 and least values of 2, by taking i successively = ^jr, — t, —a-, these quantities differing £1 £1 £* by »-, and as in the preceding integral, the first power of i only occurs, it is evident that the times of all oscillations are equal. 60 CELESTIAL MECHANICS, being the semi-circumference of a circle, of which the radius is unity ; we shall thus find > Supposing the point suspended at the extremity of a thread without mass, of which the other extremity is firmly fixed ; if the length of the thread is r, the motion of the point will be the same as in the interior of a spherical surface ; it will constitute with the thread a pendulum, of which the cosine of the greatest deviation from the vertical will be — . If we suppose that in this state, the velocity of the point is no- r thing J* it will vibrate in a vertical plane, and in this case we shall * — expressing the cosine of the angle which the radius makes with the verti- tical, when the deviation from the vertical is the greatest, z is then least, and consequently it is equal to b, .: >- is the cosine of the greatest deviation, and as generally '— =: 1 — COS. A . , . . . r — b . , . • /. 1 • ■ » , in this case it is r: to -jr— , y" — this quantity, lor making a = r in the expression for v', it becomes r-— Z^" (r—b){r+b) r—b The pendulum described in the text is merely ideal, as every body has weight. How- ever, philosophers have given a rule, by means of which we are able to determine th« length of the imaginary pendulum, such as has been described, from the compound pen- dulum which is isochronous with it. (See No. 31 of this book.) From the equation cfar. (r* — z*)=c'.(/< it follows that the angular velocity is inversly as the square of the distance; this is universally true, whenever the areas are propor- c dt tional to the times, for we have then ^•.d'a.= ddt .: d-n = -^. See note to No. 6. dx-A-dv'^ 4-dz'^ ds'' '^^ From the equation c-f-Sgz = — ■ , ■ = -r—, we derive dt == / , "■■■— '^ dt dt^ \/c+2ga, ds when the velocity — vanishes before the tangent becomes a second time horizontal. PART I.— BOOK I. 61 have, a = r; v* = The fraction is the square of the sine of half the greatest angle which the thread makes with the vertical j the entire duration Tof an oscillation of the pendulum will therefore be T=: ;■ J) If the oscillation is very small, is a very small fraction, which may be neglected, and then we shall have therefore the very small oscillations are isochronous, or of the sarrie' da- ration, whatever may be their extent ; and by means of this duration, and of the corresponding length of the pendulum, we can easily deter- mine the variations of the intensity of gravity, in different parts of the earth's surface. Let z be the height through which a body would fall by the action of gravity in the time T; by No. 10 we shall have 2z:=g T^, and conse- quently ^ = ^tt.^ r j thus we can obtain with the greatest precision, by means of the length of a pendulum which vibrates seconds, the space through which bodies descend by the action of gravity in the first se^ cond of their fall. It appears from experiments, very accurately made. ds the point describes only a part of a circle of the sphere, but if y be finite, when the tangent becomes a second time horizontal, then the point describes the entire circumference. These circumstances may be determined by means of the equation dx"- 4- di/'^+dz' 62 CELESTIAL MECHANICS, that the length of the pendulum which vibrates seconds is the same, whatever may be the substances which are made to oscillate. From which it follows that gravity acts equally on all bodies, and that it tends, in the same place, to impress on them the same velocity, in the same time. • When the oscillations are very small T = t,*/ — , and if a body vibrated in a cy- cloid whose length was equal to 2r, the time of an entire vibration would be equal to "•• f./ — , ivhatever be the amplitude of the arc, for the equation of this curve is s'^zaz. o ,_ dz , (See Lacroix Traite Elementaire, No. 102) .". dszz\/ a — =' ^"'l V 2g (A z) = V 5r — -r-j (^i equal to the value of r when ^rrO) .'. dt= — . ~—^ = —,\//^^^ \ '^^ V^i vh-7 '^ V/ (y; ^ k/ h >'' *'''~*' V V~~V arc COS. f -^^-— j + C,ifwetake this integral between the limits r=j^, z= 0,— = - . ^y/ — , ,*, if 2a=r, i, e, if the radius of the osculating circle be equal to 2a, the small oscillations in this circle are equal to the oscillations in the cycloid, and sipce /( does not occur in this integral, the time of describing all arcs of the cycloid are equal, provided one extremity of these arcs be at the lowest point. It appears from the foregoing investigation, that the time of vibration in a cycloidal arc, is the hmit to vhich the time in a circular arc approaches, when the latter becomes inde- finitely small. W hen great accuracy is required, all the terms after the two first in the series expressing the time in a circular arch are rejected, and then the expression for T'= V. w — S 1-j- f-^y (-n~)'' [ '^•"" which it appears that the aberration from isochronism varies, as the square of the sine of half the amplitude. We might determine the time of describing any given arc of a circle, if we knew the coordinates a and b, and also z the coordinate of the extremity of the arc required, for then the angle 6 w ould be determined. We might also, derive a general expression for the time of describing any given arc of a ci/cloid. For if in the initial velocity be such, as w ould be acquired in falling down a height equal to A, we shall have at any point in the ds — __— cycloid «»= 2g. {IHh—z) consequently — = \/'^g(,H-\-h — 2) .♦. dt = ===== (by substituting for ds its value /T"(-- \ PART I.— BOOK I. G3 12. The isochronism of the oscillations of the penduhnn, being only an approximation ; it is interesting to know the curve on which a heavy body ought to move, in order to arrive at the point where the motion ceases, in the same time, whatever may be the arc which it shall have described from the lowest point. But to solve this problem in the most general manner, we will suppose, conformably to what has place in nature, that the point moves in a resisting medium. Let s repre- sent the arc described from the lowest point of the curve ; z the vertical abscissa reckoned from this point ; dt the element of the time, and <r the gravity. The retarding force along the arc of the curve will be, v/ — .arc. COS. =-Tr„-r7\ \- C; we determine C by making^ = 0, and* ^ //, Qg \{H-]-h) we might deduce from this general expression, the time of describing the whole cycloidal arch; Cis equal to =Y/ — - -! arc- j cos. =-7 — — , .•. when the initial velocity vanishee C = 0, for then H vanishes. In the precedirig investigations tlie motions are supposed to be performed in a nonresist' ing medium, but this is not essentially necessary, in order that the oscillations should be iso- chronous in the cycloid, or nearly so in the circle. For it is proved in No. 12, that the os- cillations of a body moving in a medium, of which the resistance is as tiie velocity, are iso- chronous when the curve described is a cycloid, and it has been demonstrated by M. Poisson, tljat when a body describes a small circular arch, in a medium of which the resistance varies as the square of the velocity, or as the two first powers of the velocity, the oscillations are isochronous, the analytical expression indicates that the time of describing the first arc is as much lengthened by the resistance, as the time of describing the ascending arc is dimi- nished, so that the time of the entire vibration remains the same as if the body moved in a vacuo, the amplitude of the arc perpetually lessens ; and it may be proved, that if the intervals of time are taken in arithmetic progression, the amplitudes of the arcs described decrease in geometric proportion. 64 CELESTIAL MECHANICS, 1st, the gravity resolved along the arc ds, which thus becomes equal to dz ,g. — ; 2dly, the resistance of the medium, which we will express by <p. );rf . -7- being the velocity of the point, and p. X -r— > being any function of this velocity. By No. 7 the differential of this velocity will be equal to —g. -^ <p. 5 -— f ; therefore, by making dt con- Clo C Civ -^ stant we shall have - d*s , dz , cc?5) ... Let us suppose that 0.\—{ =m.— +«. -7-, and* =4/(5') > denoting (a^j dt dt* by i}/'(s') the differential oi ^ {s') divided by rf/; and by f (s') the dif- ferential of ^'(s'") divided by ds\ we shall have ds ds' ,,, ,. di=dt- ■'-('' the equation (?) will become • Substituting for ^. (c?i) its value in the equation (i), it becomes substituting these values for -r- , and — — , we shall have * dt- dt^' A^r' /?/2 //c'* rft* ff'.rfl PART I.— BOOK I. 65 We make the term multiplied by — — , to disappear by means of the equation which gives by integrating ^{s') = \og. S(A(5'+y)^} zz-s; h and q being arbitrary quantities. By making / commence with s we shall have hq '^ = \, and if, for greater simplicity we make, hzzl, we shall have ^—c"' — 1.* K V dividing all the terms by ^'.[s) and concinnating we obtain * From the value of d^s which has been already given, we get ds'^ ds* ds and by integrating we obtain, log. ds — log. ds'-{-ns = e or log. -^y = e — m ; V -Tr= —^ J and ds = , mtegratuig again we shall nave s' + y = , V \o^. (ji.{s' +q))—ns—e or dividing both sides by n; . °^ '"^^ — ^ )=((log-(«-(*+9)»^)) n ' = « , and if — , be made equal to — ^ we obtain log. ((A. (/-f-y")) T = *. If we , 1 suppose s to commence with s, they are = to 0 at the same instant, •.• log. A.y n = 0, at this instant, and consequently /i.y » = 1, y must be equal to uniiy since n is a constant indetermined coefficient, •.• log. (i'+l)n = s = ^j.(«'), and / = c"" — 1. 66 CELESTIAL MECHANICS, c being the number whose hyperboHc logarithm is tmity ; the diffe- rential equation (/) becomes then dh' , ds' , „ dz ,^ ,., ^ = ^+"-^+"^-^-(^+^^*- By supposing s very small, we may develope the last term of this equation into a series ascending according to the powers of 5' which will be of this form, ks'+ls'' + , &c. ; i being greater than unity ; the last equation then becomes at^ (It mt This equation multiplied by c~^. (cos. y? + y/_i. sin. yt), and then in- tegrated, becomes (7 being supposed equal to w^y^- "'*_) Jl f -) r (Is' f c«-jcos. yt-\-^-\. sin. yt[. j-T^- + ("f- — rV— ^ • ^ C = — l.J^'dt. c^ ^cos. yt-\-\/—\.. sin. 7^. ( — &c.t * For since sf = C" — 1, -77 = A'i.^^ = — ;jr "^ . ,,. ■•' ■\'\^V - ds ^ ' tLc"" n.(l+s) ^^ ' K'.(l-j-/)=' , , , d's' '^^ . dz •.•the equation (I) becomes ——— + w. -7- +""^--77-' (!+*)> when ^ is veiy (IZ CtC tto small the variable part of the last temi of this equation may be expanded into a series proceeding according to the ascending powers of s', for substituting in place of s' it be- dz ... comes = — - . c""*, when *• is very small s' is also veiy small, as is evident from the equa- tion s' = c'" — 1 ".• — p = the sine of the inclination of the tangent to the horizon is ds very small, and as all the terms which occur in the expression -r— . (l-f-^')' are very small it can be developed in a series of the form given in the text. f Cos.yi+\^—\- sin. yi = c'' '^-' _ ggg Lacroix Traite Elementaire, No. 164.,) ••• by substituting c'''v~' in place of the circular function, we obtain PART I.— BOOK I. 67 By comparing separately the real and imaginary parts, we will have els' two equations by means of which we can eliminate — — ; but it will be k2 &c. If we multiply both sides of this equation by dt, and then partially integrate, we shall have (the integral of rf/c(l +yV-i)'- ^ ^,^(|+yvri) *. substituting this value of /rff'.c(-1+''v/-0f- j^ jj^^ ggj,^jjj ^^^^jj ^^f ^1,^ preceding inte- gral, and for k its value y«-4 , we obtain 4 = (-^_;»yV=i.+ v'.y.'.<f/.c(r +VV^)') __;.^,,,(| + '''^^)'-^,. If we substitute for cV %/ ^ '• its value cos. yt+ ^~, si„. .^,, and concinnate, we will obtain ^ * (cos.y*+^Z:i.sin.yO(-^+a-vV=T).0=-^/^'^'/^^(^ (COS.y<.+ y^_l. sin, yt.) &C. 6d CELESTIAL MECHANICS, sufficient to consider here the following* — <// —Cm ") c 8 . — - . sin. yt+c - . *'. 1 — . sin. yt — y. cos. yt. > cit ^ A y z= — l.Js" dt.c~^ . sin. yt — &c. the integrals of the second member being supposed to commence with t. ds Naming T the value of t at the end of the motion, when —r- vanishes, at that instant we shall have c . s'.< sin. yT — y. cos. yT. <r zz. — l-Js''. dt. c^ . sin. yt — &c. When s' is indefinitely small, the second member of this equation va- nishes, when compared with the first, and we shall have ; O zz sin. yT — y. cos. yT,* * As the imaginary parts of this equation cannot be equated \\ith the real parts, the real and imaginary parts must be compared separately, which gives two distinct ecjaations, the part of this equation which is considered, is the part which was multiplied by f Partially integrating the expression — /. fs'.c —. sin. yt.dt-^ &c. we obtain I- ^ ,. Im ^ ^ , ,. li ^ 'JL ,,•,,, . C 2 .COS.yi. S' 7T- f-C 2 • (it. COS. yt. s' f.C 2 . COi.yt. S'~^ ds , if we integrate the second term of this expression, as before, we shall have lin '^ . „ ■ Im'^ ^ m , . ,■ Imi ^ . , -i , , — jr-;- c 2 . Sin. yt. s' -\- -—:^ f.C i dt.sm. yt. s'-\--—- fc a.sm. y/. «'. as', Zy 4y ' 2y in like manner the integration of the term in this last expression, which contains dt, would give terms of the same foi-m as in the preceding integral ; consequently the value of mt — I f.s' dt.c~T. sin. y<+&c. cannot be exliibited in a finite number of terms; but if the mt preceding intervals are taken from i r: 0 to < = T, then the value of — IJs. cY. dt. sin. PART I.— BOOK I. 69 consequently tang. yT= — IL, m and as the time T is, by hypothesis independant of the arc described. yt=0, for by substituting m place of cos. yTits value — -~ sin. yT, in the terms where ds "y occurs, these terms in two succeeding expressions will be equal, and affected with contrary signs, consequently they destroy each other ; \nth lespect to those terms which are free from the sign of integration f, we may remark that they resolve themselves into two de- creasing geometric series, which are respectively of the following forms / m ,. Im'^ mt . Im* J!^ ,. „ , . „ . — . cos. yt.c 2 . s' -■ COS. yt.c~. s + — — r. COS. yt.c 2 . /', &c. ad mfinitum, y iy^ lo.y* Im . '"I Im^ ^ Im^ . ^ — "H— :• sm. yt.c. ~. «.-*■——. sin. yj.c a . s' — ^^ , . sm. yt.c 2 . «"-|-&c. ad infim'tum, Zy Hy* S^y" by summing these series they come out equal respectively to I Im y mt 2y* ">' cos. yt.c » . s'i —, sin. yt.c 2 . /«, by substituting ' 4y* ^ 4y» 7)1 for cos. yTits value — . sin. yT, the first expression becomes 2y h, l + I^il 4y- sin. y T^.c". «', which is equal to tlie second with a contrary sign, consequently 7nt it follows that whatever be the magnitude /; — ^.yi',-. dtxT". sin. yt = 0, when the integral is taken from f— 0 to t=zT. The same reasoning applies to the other terms of the series, which contain powers of s' superior to i. I being independant of /, if it is equal to nothing when s' is very small it will be al- mt ways equal to nothing ; and since neither sin. yt, nor c T change their signs from f=0, to mt t=. T, it is evident that the evanescence o{Js''.c~2. sin. yt can only arise from I being equal to nothing, in this case also the coefficients of the powers of s' greater than /'. i. e. the subseqnent terms of the series vanish. 70 CELESTIAL MECHANICS, this value of tang. yT has place for any arc whatever, therefore what- ever be the value of *', we have mt 0= /.//', dt.c~. sin. 7'/ + &c. the integral being taken from t—0 to t=T. If we suppose s' very small the second member of this equation will be reduced to its first mt term, and it can only be satisfied by making / =0 ; for the factor c~ . sin. yt, being constantly positive from /— 0 to t= T, the preceding integral is necessarily positive in this interval. Therefore the tauto- chronism is only possible on the supposition of ds' which gives for the equation of the tautochronous curve g.dz-= (1— c J 71 In a vacuum, and when the resistance is proportional to the velocity, n * Substituting for 1+i' its value C", and ds' its value n.ds.C", we obtain n'^S-dz. „.. , , _. _ , k.ds n.ds.C^ 1.2ns = ^(cnj _ J ) ._. gj^ _ J±± jl — c""" i , •.• when the body moves in a vacuo, or in a medium of wliich the resistance is proportional to the velocity, n=zo :• gdz — ' (1 — c-"*) = ks. — , but if we express c-"* in a se- nes Jt becomes = l ~-\ — — -, &c. •.• the general expression for * n \ ^1 1.2 ^ 1.2.3 ; , when 7! = 0, lc.ds.s. From this equation it follows that k = , , this is also g.p true, ds.s when n has a finite value, if s be taken very small, as is evident from the preceding series. PART I.— BOOK I. 71 is nothing ; and this equation becomes g.dzz=.hs.ds ; which is the equa- tion of the cycloid. It is remarkable * that the coefficient n of the part of the resistance, which is proportional to the square of the velocity, does not enter into the expression of the time T; and it is evident from the preceding analysis that this expression will be the same, even though we should add to the expression for the law of the resistance, which has been given above, the terms, ds' , ds* c. p. + q + &c. ' dt' ^ dt If in general, R represents the retarding force along the curve, we, shall have s being a function of t, and of the entire arc described, which conse- quently, is a function of t and of s. By differentiating this last function, we obtain a differential equation of this form, dt V being a function of / and of s, which, by the conditions of the pro- blem must vanish, when t has a determinate value, which is indepen- dant of the whole arc described. Suppose, for example, V ■=. S.T, S * Since the value of T is the same when the terms P. —H 4. n. — ^ + &c. are added rff 3 ^ " tit* ' to VI. ——■ -}- n.—— , it follows that the generality of the conclusion is not affected by ds ds^ ( ds ^ substituting m. -7-+ n. -— - in place of ip ] -7- j • 72 CELESTIAL MECHANICS, being a function of s only, and T being a function of t only ; we shall have d-s „ dS ds „ dT dS ds^ ^ ^ dT ^ = 1 . — ; — • — r— + O. = . — - + *J« , • j dt"^ ds dt dt S.ds if dt but the equation — — = ST, gives t, and consequently — — equal to a dt ^ at function of -, which function we will denote by -—-, 4' » tttt i therefore we shall have d's ds' { dS ,1 ds dt^ S.dt^ {f + *(^)} = -^- Such is the expression for the resistance which corresponds to the ds differential equation — = ST; and it is easy to perceive that it involves the case of the resistance proportional to the two first powers of the velocity, multiplied respectively by constant coefficients. Other differential equations would give different laws of resistance.! * S being a function of x, which is a function of t, tlie differential coeflacient of S, with resDect to ^ = — '— '—r > and substituting for T its value ^ , ■ we obtain •^ ds dt ii-as d^s dS ds"^ „ dT dt^ ~ S.ds dt' ' dt ' f In the precedhng investigation the body is supposed to ascend from the lowest point, and the curve which then satisfies the condition of tautoclironism is U7iiqtte in a given medium ; but if the body descended from the highest point, then it would oscillate at the other side of the point where the tangent was horizontal, and the problem becomes somewhat more indeterminate, in this case k may be announced more generally thus ; to find the lines, the time of describing which will be given, whatever be the amplitude of the arcli described ; the discussion of this problem is too long to be inserted here, the reader will find a complete investigation of it by Euler in the Transactions of the Academy of Petersburgh for the years 1764 and ITS*, he demonstrates tiiat the arcs at each side •f the lowest poijit are not necessarily equal and similar, however, the sum of these arcs PART I.— BOOK I. 7S is proportional to the square root of the vertical coordinate, •.• the curve whose length is equal to the sum of these arcs will be the common cycloid, in like manner, if we have the differential equation of one of these arcs, we can determine the differential equation of the other ; if the first arc be a cycloid, the second will also be the arc of a cycloid : in this case the time of describing each of the cycloidal arcs will be constant, howevef the generating circle of the second cycloid is not necessarily equal to that of the first. If we combine the condition of tautochronism, with the condition of the two branches at each side of the lowest point, being equal and similar, the curve will be then the vulgar cycloid, therefore this is the only plane curve in which the sum of the times of the ascent and descent is always the same in a vacuo ; but this property belongs to an indefinite number of curves of double curvature which are formed by applying the cycloid to a vertical cylinder of any base, the altitude of the curve above the horizon remaining the same as before, for «* ds^ + ds =: -— = c — 2gz, '.' dt= — , consequently the value of t depends on the initial '^** \/c—2gz velocity, and on the relation between the vertical ordinates and arc of the curve •.' what- ever changes are made in the curve compatible with the continuity, the value of dt will not be changed, provided the preceding relation remains ; and it follows conversel}', that the projection of any tautochronous curve of double curvature, on a vertical plane, will be a cycloid with a horizontal base. In the cycloid, if a body falls freely, the accelerating force along the tangent varies as the distance from the lowest point, for 4*=4a2, '•' g--f- (= accelerating force =: -^ , I Us ^(t J the pressure arising from gravity = g. — — , and the pressure which is produced by the centntugai torce — - for radius of curvature = I.kj a{a — 2), and the square of 2.\/(i(a — 2) V \ / 1 the velocity = 'i-g-{a — r), see No. 9, (the coordinates of z are reckoned from the lowest point ;) it follows from the preceding expression that the ivhole pressure at the lowest point, and consequently the tension at this point of a body vibrating in a cycloid is r= to twice the gravity. When a body describes a cycloid, the accelerating force varies as the distance from the lowest point, as has been stated above ; and if a body was solicited by a force varying according to this law, the time of falling to the centre will be given, for we have ^ = — As V '^^2= — ^-5 ■^. ••• "' = — -^«* + C, v=0, s = S, V Cz=AS', ••• dJt "i at v=A. ^ s«_ii &^ •*= -7-. ^, V ^*/ = arc. cos.^, and when s—O, t=T V^ O S* i 74 CELESTIAL MECHANICS, 32,—^ , consequently the time of descent to the centre, is the same from whatever point, the body begins to fall. From the preceding expression, it follows, that the time of de- scribing any space s, varies as the arc, and the velocity acquired varies as the right sine. Se« Princip. Mat. Prop. 38, Book 1st. PART I.— BOOK I. IS CHAPTER III. Of the equilibrium of a system of bodies. 13. The simplest case of the equilibrium of several bodies, is that of two material points meeting each other with equal and directly con- trary velocities ; their mutual impenetrability evidently annihilates their motion, and reduces them to a state of rest. Let us now suppose a number m of contiguous material points, arranged in a right line, and moving in its direction with the velocity u, and also another number ?«' of contiguous points, disposed in the same line, and moving with the velocity u', directly contrary to u, so that the two systems may strike each other ; there must exist a certain relation between u and u', when both the systems remain at rest after the shock. In order to determine this condition, it may be observed that the system m, moving with the velocity u, will constitute an equilibrium with a single material point, moving in a contrary direction with the velocity mu ; for every point of the system would destroy in this last point, a velocity equal to u, and consequently the m points would destroy the whole velocity mu ; we may therefore substitute for this system a single point, moving with the velocity mu. In like manner we may substitute for the system m', a single point moving with the velocity m'u' ; now* the two systems being supposed to constitute an equilibrium, the two points which are substituted in their place, ought to be also in equilibrio, therefore their velocities must be equal j consequently we L 2 * These two systems of contiguous material points, may be supposed to represent two bodies M, M', of different masses, equal respectively to the sum of all the ms, and in',s. 76 CELESTIAL MECHANICS, have for the condition of the equilibrium of the two systems, mu-^m'u'. The mass of a body is the number of its material points, and the product of the mass by the velocity, is what is termed its quantity of motion ; this is also what we understand by the force of a body in motion. In order that the two bodies, or two systems of points whicli strike each in contrary directions, may be in equilibrio, the quantities of motion or the opposite forces must be equal, and consequently the ve- locities must be inversely as the masses. The density of bodies depends on the number of material points which tliey contain in a given volume. In order to determine their ab- solute density, we should compare their masses with that of a body t which has no pores ; but as we know no such body, we can only deter- mine the relative density of bodies, that is to say, tlie ratio of their density, to that of a given substance. It is evident that the mass is in the ratio of the volume and density ; therefore, if we denote the mass of the body by M^ its volume by C/", and its density by D, we shall have generally M= U. D ; in this equation the quantities AI,D,U, relate to the units of their respective species. In what precedes, we suppose that bodies are composed of similar material points, and that they only differ in the relative situation of these points. But the intimate nature of matter being unknown, this supposition is at least very precarious, and it is possible that there may be essential differences^ in their integrant molecules. Fortunately, the truth of this hypothesis is of no consequence to the sci- ence of mechanics, and we may adopt it without any apprehension of 7 Distilled water, at its greatest density, is the substance which has been selected for the term of comparison, as being one of tlie most homogeneous substances, and tliat which may be readily reduced to a pure state. X By the integrant molecules of bodies, as contradistinguished from their constituent parts, we understand those which arise from the subdivision of the body, into minuter por- tions ; by the constituent parts are understood the elementary substances of which a body it composed. PART I— BOOK I. 77 error, provided that by similar material points, we understand points which, when they meet with equal and opposite velocities, mutually con- stitute equilibrium, whatever their nature may be.* 14. Two material points, of which the masses are m and »/, can only act on each other in the direction of the line joining them. Indeed, if the two points are connected by a thread passing over a fixed pully, their reciprocal action cannot be directed along this line ; but the fixed j)ully may be considered as having at its centre a mass of infinite den- sity, which reacts on the two bodies, so that their mutual action may be considered as indirect. Let p denote the action which is exerted by in on rri by means of the right line which joins them, which line we suppose to be inflexible and without mass. Conceive this line to be actuated by two equal and op- posite forces p and — p ; the force — p will destroy in the body m a force equal top, and the force/? of the right line will be communicated entirely to the body rri. This loss of force in m, occasioned by its action on m', is termed the reaction of m' ; therefore in the communication of motions, the reaction is ahvays equal and contrary to the action. It appears from observation that this principle obtains for all the forces of nature.! * It' there be actually essential differences in the integrant molecules, then it is noc inconsistent to suppose, with some philosophers, that the planetary regions are filled with a very subtle fluid destitute of pores, and of such a nature as not to oppose any resist- ance to the motions of the planets. We can thus reconcile the permanency of these motions, which is evinced by observation, with the opinion of those philosophers who regard a vacuum as an impossibility ; however the plenum, for which De-Cartes contended, is not confirmed by the preceding hypothesis, as he held that all matter was homogeneous, and that the ether, which, according to him filled the planetary regions, differed from other substances only in the form of the matter. See Princip, Math. Book 2, Prop. 4-0 ; Exper. l*, and Book 3, Prop. 6, Cor. 2 and 3 ; Newton's Optics, Queiy 18; and Systeme de Monde, page 166. However, as extension and motion are the only properties which are taken into accoimt in Mechanics, it is indifferent whether matter be considered as ho- mogeneous or not. f This equahty does not suppose any particular force inherent in matter, it follows ne- cessarily fi-om this, that a body cannot be moved by another body, without depriving this body of the quantity of motion which is acquired by the first body, in the same manner as when two vessels communicate with each other, one cannot be filled but at the expense of the other. 78 CELESTIAL MECHANICS, Let us suppose two heavy bodies m and m' attached to the extremities of an horizontal right line, supposed to be inflexible and without mass, which can turn freely about a point assumed in this right line. In order to conceive the action of those bodies on each other, when they are in equilibrio, we must suppose the right line to be bent by an indefinitely small quantity at the assumed point, so as to be formed of two right lines, constituting at this point an angle, which differs from two right angles by an indefinitely small quantity w. Let J' andj' represent the distances of m and m' from the fixed point ; if we resolve the weight of m into two forces, one acting on the fixed point, and the other directed towards nz', this last force will be represented by ^ , g being the force of gravity. In like manner the action of ni on m will be re- presented by — '^ . , the two bodies constituting an equilibrium, these two expressions will be equal, consequently we will have 7n/=m'J" ; this gives the known law of the equilibrium of the lever, and at the same time, enables us to conceive the reciprocal action of pa- rallel forces. Let us now consider the equilibrium of a system of pointsactuated by any forces whatever, and i-eacting on each other. Let^representthe distance of m from ni \f' the distance of m from m\f" the distance of wj' from /«", &c. * Gravity must be distinguished from weight ; the weight of a bedy is the product of the gravity of a single particle^ by the number of particles. If we conceive a line drawn from the fixed point, parallel to the direction of gravity, meet- ing a line connecting ni and ?»', this last line will be q.'p., horizontal, and therefore perpen- dicular to the vertical line, which will *.• be equal toy multiplied into the sine of the angle whichy makes with the horizontal line, but as the sides are as the sines of the opposite angles, we liave the sine of the angle whichy makes with the horizontal line, to the sine of u, or its supplement, as, f':f-{-J' •.■ it is equal to :i-^ — j^:=q.p.-4——-,,nowi£tiie weight be represented by the vertical line, then mg divided by sine of the angle whichy makes with the horizontal line, i. e, — '^ ., ■ will be the force in the direction of/. "J PART I.~BOOK I. 79 also let p be the reciprocal action of mon m' ; p' that ofm on m'^ ; p'' that of m' on m", &c. and lastly, let mS, m'S', rri'S", be the forces which act on m, rri, m'' ; &c. 5, /, s", lines drawn from any fixed points in the di- rection of these forces, to the bodies m, m', rd\ Sec. ; this being premised, we may consider the point m as perfectly free, and in equilibrio in con- sequence of the action of the force mS, and of the forces, which the bodies m, ni, m\ communicate to it ; but if it was subjected to move on a curve or on a surface, it would be necessary to add to these forces, the reaction of the curve or of the surface. Therefore, let Ss be the varia- tion of s, and let S^ f, denote the variation of y, taken on the supposition that rri is fixed. In like manner let S^f, be the variation of ^', on the supposition that iti' is fixed, &c. Let i?, B!, represent the reactions of the two surfaces, which form by their intersection the curve on which the point is constrained to move, and let J'r, Sr' be the variations of the di- rections of these last forces. The equation {d) of No. 3, will give : Qz=.mS.Ss +p.S,f-\- p'JJ'+kc. + mr + R'Sr' + &c. In the same manner m' may be considered as a point perfectly free, re- tained in equilibrio by means of the force niS', of the actions of the bodies m, rri, iri', and of the reactions of the surfaces on which ni is constrained to move, which reactions we will denote by R", and R". Let, therefore, the variation of s' be called Ss', and the variations of^ and/^', taken on the supposition that m and m" are fixed, be respectively S,,f, J,y""; in like manner, let Sr", Sr'", be the respective variations of the directions of R", R'", and we shall have for the equilibrium of ni O = m'S, Ss' +pJ„f+ f.SJ" + &c. + R'.Sr^' + R'aJr"'. If we form similar equations relative to the equilibrium of m'', and ml", &c. by adding them together, and observing that ifv=i,f\i„f;iS' = lf^S,f',* &c. Sf, and Sf\ being the total •3/"=3^/'+3„/; 3/"' - 3,/'+S„/'/ + &c. ; for/ and y are respectively functions of the coordinates of their extreme points, and when these are moved by an indefinitely fmall quantity, all the powers of the increments of the coordinates, after the first may be rejected, and then the entire increment of _/ is equal to the sum of the partial incrementa 80 CELESTIAL MECHANICS, variations ofyandy'+&c. we shall have 0=-z.m.SJs + xp.if-^j:RJr; (k) in this equation, the variations of the coordinates of the different point? of the system are entirely arbitrary. It should be observed here, that in consequence of the equation («) of No. 2, we may substitute in place of mS.Ss, the sum of the products of all the partial forces by which m is actuated, multiplied by the respective variations of their directions. The same may be observed of the products m'Sis' ; If the distances of the bodies from each other be invariable, i. e. if J^y',jr'''', + &c. are constant, this condition may be expressed by making {/=0, Sf' = 0, &c. The variations of the coordinates in the equation (k) being arbitrary, they may be subjected to satisfy these last equations, and then the forces p, p',p'', &c. which depend on the reciprocal action of the bodies composing tl:e system, will disappear from this equation ; we can also make the terms li.Sr, R'Ji'. •+ &c. t to disap- pear, by subjecting the variations of the coordinates to satisfy the equa- tions of the surfaces, on which the body is constrained to move. The equation (/.) will then become 0=T.mS.Ss; (I) from which it follows that in case of equilibrium, the sum of the varia- which are due to the separate variation of each coordinate, ••■ the entire variation of y is equal to the sura of the partial variations, which correspond to the characteristics 3, and i^. * From this it appears,' that the conditions of the equilibrium of a system of bodies, may be always determined by the law of the composition of forces ; for we can conceive the force by which each point is actuated to be applied to the point in its direction, where all the forces concurring, constitute an equilibrium when the point is entirely free, or which constitute a resultant, which is destroyed by the fixed points of the system, when the point is not altogether free, f See Notes to No. 3. Tlie equation (/) obtains, whether the points are all free, or are subjected to move o* PART L— BOOK I. 81 tions of the products of the forces, into the elementary variations of their directions will be equal to nothing, whatever changes be made in the position of tlie system compatible with the conditions of the con- nection of the parts of the system . We have arrived at this theorem, on the particular supposition of the parts of the system being at invariable distances from each other ; how- ever it is true whatever may be the conditions of the connection of the parts of the system. In order to prove this, it will be sufficient to shew that when the variations, of the coordinates, are subjected to those con- ditions, we have in the equation (Z,) 0 = I..p.Sf-^I..R.h- ; but it is evident that Sr, Si^, &c. are equal to nothing, when these con- ditions are satisfied ; therefore it is only necessary to prove that in the same circumstances we liave 0 = i:.p.Sf. Let us therefore suppose the system actuated by the sole forces j9, pf, p, &c. and that the bodies are subjected to move on the curves, which they can describe in consequence of the same conditions ; these forces may be resolved into others, some of which q, q', q", &c. acting in the direction of J,' J', f", &c. will mutually destroy each other, without producihg any action on the curves described ; others will be perpendicular to those curves ; and others again will act in the direction of tangents to those curves, by the action of which the bodies may be moved ; but it is easy to perceive that the sum of these last forces ought to be equal to nothing ; since the system being by hypothe- sis at liberty to move in their directions, they are not able to produce either pressure on the curves described, or reaction between the bodies ; M curved smfaces ; in the former case, the forces S, S', S", constitute an equilibrium ; in the latter case, these forces have a resultant, of which the direction is perpendicular to the surface. (See Note to page 17.) 0*- ro 8J CELESTIAL MECHANICS, consequently they cannot constitute an equilibrium with the forces — P' — P'f — F^'t ^^' y> ?'» I"y ^^- T, T', T" ; therefore they must vanish, and the system must be in equilibrio in consequence of the sole forces py—p',—p", &c. ; q, q', q", &c. ; T, T', &c. Now, if Si, Si', &c. repre- sent the variations of the directions of the forces T, T', Sec. we shall have in consequence of the equation (A-) 0 = l.(q—p)Jf + i:.TJi ; but the system being supposed to be at rest, in consequence of the sole action of the forces q, q', &cc, without any action being produced on the curves described, the equation (/c) gives us also 0 — l.qJJ';* conse- quently we have 0 = ■s:.pJf—I,.TJi ; but as the variations of the coordinates are subjected to satisfy the con- ditions of the curves described, we have Si, = 0, Si', = 0, &c. ; therefore the preceding equation becomes O =. l.p.Sf;f as the curves described are themselves arbitrary, and are only subjected to the conditions of the connection of the system, the preceding equa- tion obtains, provided that we satisfy these conditions, and then the equation (k) will be changed into the equation (/). The following principle, known by the name of the principle of virtual velocities, when analytically expressed, is represented by this equation. It is thus an- * 0:= 2 y.S/, for q, q, q", are directed along the lines yjy'.y""; and are supposed to destroy each other without producing any action on the curves described. f The object of the second part of this demonstration is to shew, that if the system is at rest, and acted on by the sole forces /), //, ■p"-, these forces may be so decomposed as to afford forces equivalent to the reciprocal actions of the respective bodies, and that the remaining portions of the forces, as well as these reciprocal actions, will balance each other, in case of etjuilibrium, according to the terms of the proposition. Since the equation {k\ is reduced to the equation (/), when we subject the variations of tlie coordinates to satisfy the equations of the surfaces, on which the bodies are con- strained to move, it follows that it is not necessary to compute the forces f, p, &c. in order to derive the equations of equilibrium in each particular case. PART I.— BOOK I. 83 nounced: " If we make an indefinitely small variation in the position* of a system of bodies, which are subjected to the conditions they ought to fulfil, the sum of the forces which solicit it, multiplied respectively by the space that the body to which it is applied, moves along its direction, should be equal to nothing in the case of the equilibrium of the system." This principle not only obtains in the case of equilibrium, but it also insures its existence. Let us suppose, in fact, that whilst the equa- tion (0 obtains, the points m, m', &c. acquire the velocities v, v', in consequence of the action of the forces mS, m'S', which are applied to them. The system will be in equilibrio in consequence of the action of these forces, and of — 7nv, — m'x/, &c. ; denoting by Sv, ix/, &c. the variations of the directions of these new forces, we shall have in con- sequence o£ the principle of virtual velocities 0 = l.mS.SS'—^.mvJv, but by hypothesis l.mS.Ss.zzO, therefore we have 0=l.mv.5v. We may suppose the variations Sv, Sx/, &c. equal to v.dt, i/dt, &c. since they are necessarily subjected to the conditions of the system, and then we have 0 = I,.mv\ and consequently v=0, v' = 0, &c. that is to say, the system is in equilibrio in consequence of the sole forces mS, m',S', &c. The conditions of the connection of the parts of the system may be always reduced to equations between the coordinates of the several bo- dies. Let M = 0, m' = 0, &c. be these different equations, by No. 3, we can add to the equation (Z), the function xSu, x'Suf, &c. or l\hi ; \, x', being indeterminate functions of the coordinates of the bodies, the m2 * When an indefinitely small change is made in the position of the system, so that the conditions of the connections of the points of the system may be preserved, each point advances in the direction of the force which solicits it by a quantity equal to a part of this direction, contained between the first position of this point, and a perpendicular deraitted from the second position on this direction ; these indefinitely small hnes are termed the virtual velocities ; they have been denominated vertual, because the system being in equilibrio, these changes may obtain without the equilibrium being disturbed. 84 CELESTIAL MECHANICS, equation will then become 0 = I..mS.Ss-\-I,.xSu ;* in this case the variations of all the coordinates are arbitrary, and we may equal their coefficients to nothing ; which will give as many equations, by means of which we can determine the functions x,x'. If we com- pare this equation with the equation {k) we shall have l.xJu = l.pJf+l.RJr; by means of which we can easily determine the reciprocal actions of the bodies m, m', &c. on each other, and also the forces — R, — R', with which they press against the surfaces on which they are constrained to move. 15. If all the bodies of the system are firmly united to each other, its position will be determined by that of three of its points which are not in the same right line ; the position of ' each of these points de- pends on three coordinates ; this produces nine indeterminate quan- tities ; but we can reduce them to six others, because the mutual dis- tances of the three points are given and invariable ; these being sub- stituted in the equation (/)» will introduce six arbitrary variations ; by supposing their coefficients to vanish, we shall obtain six equations, which will contain all the conditions of the equilibrium of the system : let us proceed to develope these equations.! * By means of the formulae which are given in the notes to No. S, page H and 15, we can determine A, >! , &c. when S, S^, S*, are given for each individual point ; and there- fore ;;, p, p", k, k', ¥, by means of the equation 2.A. Jie = l.p^f-^- 'S.R.^r ; in the equa- tion Z ?n.SJ«-f- 2. a.5m, m, m', m", &c. may be considered as entirely free ; and if we put the coefficients of the variation of each variable equal to nothing, and then eliminate the indeterminate quantities, A, /', A-^, &c. between these equations, the expressions which re- sult, will give the relations which must exist bstween S, Sf, S", &c. and the coordinates, when the system is in equilibrio. f It follows immediately, from the demonstration of the principle of virtual velocities, that it has place for all the indefinitely small motions which can be given to a solid body, which is either free or constrained to certain conditions, for in all these motions the re- spective distances of the points of the body remain the same. PART I.— BOOK I. 85 For this purpose, let x, y, z, be the coordinates of m ; x', i/, ^, those of m' ; x", y", z", those of m''^ ; &c. ; we shall have then f'= x/{^'^—x)*+(i/"—i/)*+iz'-zy and if we suppose ix = jy = sjo" = &c. $y = Sy = Sy" — &c. ; Sz — Sz' = Sz'' = &c. ; we shall have $f= 0, Sf'= 0, Sf^^=^ O, &c. j* the required condi- tions will therefore be satisfied, and from the equation (/) we may infer we have thus obtained three of the six equations, which contain the conditions of the equilbrium of the system. The second members of these equations are the sum of the forces of the system, resolved pa- rallel to the three axes of x, y, and z, therefore each of these sums must vanish in the case of equilibrium. And as the number of the equations of equilibrium, which are derived from the principle of virtual velocities, is always equal to the number of possible motions, this number being equal to six, in the case of a solid body, or of a body whose parts are invariably connected, the number of equations of equilibrium will be six in like manner. consequently when 3x'=5x, 3j/'=;3y, Sz'=h, &c. Sy=0, therefore 2m.S. -j — | = 0, 2^.S, S ^ I = 0. &c. ; for when ix=i3f - Jx" ; iy—^Z^y ; 3^ == ?-' = ^^'= ^'^- • 80 CELESTIAL MECHANICS, The equations Sf^ =0, if' = 0, $/"== 0, &c. will be also satisfied, if we suppose, z, z', z", invariable, and then make Sx = ySw ; Sy = — x.Jw ; Saf — yi.iu, &c. iy == — x'.Su, &c. SiAt being any variation whatever. By substituting these values in the equation (/)» we shall have .0 = ..„*.|^.(|)_,.(|)|. It is evident that we may, in this equation, change either the coor- dinates x, x', x", &c. or the coordinates y, y', y", &c, into z, z\ z", which will give two other equations, and these reunited with the pre- ceding equation, will constitute the following system of equations : «=x»,5.^^.(A)_,.(|.)^i 2i».S.3*=:0, is equivalent to SmS. S -i \ .S^ — 0, 2mS. ^ -j^ | . J^ = 0, 2.OT.S. -! y- f -'^ :;^0. See Note to No. 2, page 9. * In like manner, if we suppose, 3x=y.3«r, 'ixzzy'^n, Sy = — «?«, 3y:= — x'i», 'if, If, &c. = 0, for substituting in the preceding expression for 'if, which has been given, for >x, ix', iy, 'iy, and it becomes ^ (xW).0,'-y) + (y-3,).(x-xO ^^^ ^^^ j^^p^ By substituting in the equation, lm,Sis:z.O, for 2x, iy, &c. their values it becomes -M» -11 -'{-¥}'-=»■ When all the forces are applied to the same point, the three first equations suffice for the equilibrium ; but when these forces act in different points of space, or when they are PART I.— BOOK I. 87 by No. 3, the function smSi/.] — i is the sum of the moments of all the forces, parallel to the axes of x, which would cause the system to c is ^ revolve about the axis of z. In Hke manner, the function ^m.S.r.) ■—[ is the sum of the moments of all the forces parallel to the axes of i/, which would cause the system to revolve round the axis of z, but in a direction contrary to that of the former forces ; therefore the first of the equations CnJ indicates that the sum of the moments of the forces is nothing with respect to the axis of z. The second and third equations indicate, in a similar manner, that the sum of the moments of the forces is nothing with respect to the axes of ^ and x, respectively. If we com- bine these three conditions with those, in which the sum of the forces pa- rallel to those axes, was nothing with respect to each of them ; we shall have the six conditions of the equilibrium of a system of bodies inva- riably connected together.* If the origin of the coordinates is fixed, and firmly attached to the system, it will destroy the forces parallel to the three axes, and the conditions of the equilibrium of the system about this origin, will be reduced to the following, that the sum of the moments of the forces which would make it turn about the three axes, be equal to nothing, with respect to each of them. t applied to different parts of the same solid body, it is also requisite that the moments of the forces with respect to axis of x, y, and z, should be respectively equal to nothing. * If all the points exist in the plane of x, y, then 3z, J2', Si", are equal respectively to nothing, consequently the equations of equilibrium are reduced to the three following : ..,.s.{^| =0, ...-...{^jro. -..s.{,.(|)} -^.(l-j} f When the origin of the coordinates is fixed and invariably attached to the system, the number of possible motions is reduced to three, therefore the number of equations of equilibrium will be three ; this also appears from considering that the number of inde- terminate quantities may be reduced to three, because the distances of any three assumed points in the system, not existing in the same right line, from the fixed origin of tht coordinates, are given. $8 CELESTIAL MECHANICS, f In this case, the resultant of all the forces which act on the body passes through the fixed pomt, which resultant is therefore destroyed by the resistance of the fixed point, and it expresses the force with which this point is pressed. (See last note to No. 3.) WTien there are two points of the system fixed and invariable, then the only possible motion, which can be impressed on the body, is that of rotation, about the line joining the given points, consequently if this line be taken for the axis of z, there will be but one equation of equilibrium, i. e. ^.mS. ^ i/. (-5—) — x. ( -r—) \ = 0, this is also manifest from the circumstances of the indeterminate quantities, wliich were six in number when there was no fixed point, being reducible to one, when the origin of the coordinates, and also another point of the system, were fixed and invariable. The forces parallel to the axes of z cannot produce any motion in the s)'stem, ".' it is only necessary to consider those which exist in the plane of x, ^ ; and as to those, it is evident, fi'om the equation 2m. S. -' „. (-r-) — X. f J-) j- = 0; that their resultant passes through the origin of the coordinates, its direction will be perpendicular to the axis of z, and its intensity will express the force with which it presses on this axis. When the number of fixed points is three, there is evidently no equation of equilibrium. If the forces S, S', S'', &c. do not constitute an equilibrium, in order to reduce them to the least possible number, we should resolve them into three systems of forces, parallel respectively to the axes of x, oiy, and of ;:, then reducing the forces parallel to the axes of X, and ^, to forces — to them respectively, but acting in the same plane, which is always possible, if this last system of forces, and also the forces parallel to the axis of z, have separately unique resultants ; and if these resultants exist in the same plane, we can compose them into one sole force, which will be the resultant of the given forces ; but if the forces directed in the plane x, y, can only be reduced to two parallel forces, not re- ducible into one, then if" we combine them with the force parallel to the axis of z, the en- tire system of forces, will be reduced to two parallel ones acting in different planes, conse- quently irreducible into a unique force. Denoting S.m.S. |-^l ; S.m.S. |.^| ; 2.)n.S. | -i- i, by P, Q, R; respectively, and -X. fii}]. 2...S. |,.]^} -=.{^}}, by L, M,N.,,x„ ,, be the coordinates of that point in which the resultant of all the forces meets the plane of the axes of x, y, we shall have by the last note to No. 3, P.y,, — Qx^, n L ; Ra^., M N = M; — Q.y^, — N; therefore x., = — tt-; v., =: — „ > substituting thtse ex- R R pressions for x^ and y^, In the equation P-y,, — Q-x/, =^ L, we will obtain the equa- PART I.— BOOK I. 89 •tion L.R+M.Q-{-N.P^O, which may be considered as an equation of condition which must be satisfied, when the forces which act on the different points of the system, have an unique resultant. We must however except the case where P, Q, R, are res- pectively equal to nothing; for then the forces are reducible to two parallel forces zz., but not directly opposed to each other. If only P, and Q vanish, then in order that the preceding equation may be satisfied, it is necessary that L should vanish, consequently since P, Q, and L vanish, the forces which are directed in the plane, .r, y, constitute an equilibrium, •.• the unique resultant of the forces S, S', S", &c. must be the same with the resultant R, of the forces parallel to the axes of z, •.' we conclude that if L does not vanish when P and Q vanish, the forces have not an unique resultant, since the forces in the plane of x, y, are in this case evidently irreducible to one sole force ; if however only one of the three sums P, Q, R, vanish, then the forces in the plane .r, i/, and those parallel to the axes of z, would have respectively unique resultants, consequently the pre- ceding equation of condition would apply to this case. When the forces have an unique resultant, it is very easy to determine its position with respect to the coordinates, for if we denote this resultant by V, we shall have V^ ^ P'-\-Q^-\- R-, and—jT-, —, — = the cosines of the angles which V makes with the axes of x, y, and z, respectively, and — — , — , are the distances of the in- H R tersection of V with the plane of x, y, trom the axes of j: and y, respectively. Supposing the system to revolve round the axis of z, the elementary varia- tions of X and y, Sic. are r: respectively to y'^a, — rSa ; if y be made the axis of rotation, and J<p the variation of the angle, then we shall have 5x — — z.'^p ;Jz = -\- x.'i^ ; in like manner, x being the axis of rotation, and 34- the corresponding va- riation of the angle, «!y = -^-z.l^ ; iz = — y,'^^ ; &c. ; now if the three rotations be sup- posed to take place together, we shall have the entire variation of x=y.Su — z.Jip, of y := z.i-^ — x.^ai of z = x-isp — y-i^^, and similar expressions may be derived for the va- riations of x', y, z', x", &c. ; now if we substitute these values for Jx, ^y,-i- &c. in the equation flj, we shall have the equation L2<p-^- M.J-J/, A'.S<a=:0, L, M, N, indicating the same quantities as before ; this equation is evidently equivalent to the equation (?i) ; when the coordinates x, y, z, of any point of the system are proportional to the elementaiy variations i^, S.p, S«, ;: ^^=y.Su, z.Sij' = ^^^> x2p z=.y'^^. And consequently Sa- := 0, Sy =^ 0, Jz ^ 0 ; '." this point and all others which have the same property are immoveable, during the instant the point describes the angles 3^, ^, S«, by turning round the axes of x, y, and z ; all points possessing this property exist in a right line passing through the origin of the coordinates, see No. 28, as the cosines of the angles m, n, I, which this hne make with the axes of x, y, and 2, are ■= in this case / r in mis waBc — ^-===;;^^=:= V S« t- -i:t-+ d± -i . .y _ C Sf } N 90 CELESTIAL MECAHNJCS, Let us suppose that the bodies m, m', m", are subject to the sole force S ii I :• the right line which makes with the axes, angles whose cosines are equal to those ex- pressions, is the locus of all the points, which are quiescent during the instantaneous ro- tation of the system. Making S«=-y/3<P+5^'-|-S«*, we obtain i4> = S«. cos. m\ ^p ~ J*. cos. n ; S« = S«. cos. ^; consequently Sx = {y. cos. l—z. cos. w). Sfl; ij/ = (z. cos. m — X. cos. /.) ^6 ; Ss. = (x. COS. 71 — 7/. cos. m.) ^S, substituting for Sx, Si/, h, these values in the expression Sx* + Sy*-)"^2^> which is equal to the indefinitely small space described by the point whose coordinates are x, y, z, and observing that cos.* / + cos.' j«-)-cos.^ n= 1, it becomes equal to {x*-\-i/^+z^ — (x. cos. m-\-y cos. }i-\-z. cos. l.y). Ss^ x. cos. l-\-y. cos. m+z. cos. n. is proportional to the cosine of the angle which the line whose coordi- nates are x, i/, z, makes with the right line which makes the angles /, in, n, with the axes of X, y, z, '.' when the line drawn from the origin of the coordinates to the point whose coordinates are x, y, z, is perpendicular, to the instantaneous axis of rotation, the elementary space described by a point so circumstanced — ^x»-)-^* -f-z-. Ss, tliis agrees with what is demonstrated in No. 28. If we suppose d^, 2p, Sv, proportional to Z,, A/, A^, and make H = ^ L^ + M'+N', then L S^)/ M S(p N iu , ~7y = -rr = COS. m ; ——-= -r— = cos. n. -— = -r— = COS I. H Ss H Si H Se ■•• 1' = H.cosm; M = H. cos. n; N = H. cos. I; ••• if // = i, m = 0, n = 100°, / = 100° ; •.• L, the moment of the force is a maximum when = H, and the moments whose axes are perpendicular to the axis of H, will be equal to nothing. Tliis will be more •fially explained in Nos. 21, and 28, it is mentioned here in order to shew how the conditions of the equilibrium of a solid body may be expressed by means of the greatest momt?nt, and unique resultant; if this resultant, and this moment respectively vanish,then ij»=0, H=0, i. e. P,Q,R ; L,M,N, which are equivalent to the equations (j») (re), are equal re4)ectively to nothing ; consequently the evanescence of H and R contains the six equations of the equilibrium of a system, whose parts are invariably connected ; and as by No. 3, the sum of the moments of the composing forces with respect to an axis, is equal to the moment of the projection of the resultant of these forces ; this resultant must necessarily exist in that plane, in which the moment is the greatest possible, •.• the perpendicular to tliis plane L M N must be at right angles to the resultant, consequently, as — — , —ry, -77 , are equal to H H H the cosines of the angles which the axis of the greatest moment make with the axis of P Q It X, y, and z, and as -rrr-, — , — , are equal to the cosines of the angles which V, the unique resultant makes with the same axes ; by note to No. 2, page 7, we have LR+MQ + NP=^0, wliich is the equation indicating that the forces have an unique resultant PART I.— BOOK I. 91 of gravity, as its acts equally on all bodies ; and as we may con- ceive, tiiat its direction is the same, for all the bodies of the system, we shall have S, =S', = S^=8cc.; whatever may be supposed the direction of s, or of the gravity, we shall satisfy the thi'ee equations C^J' by means of the three following :* O = S.m.-y ; O = l,.m.i/ ; 0 = ^.m.z ; Co J N 2 • The force of gravity being uniform, and the direction of its action being always the sa™e,5=S'=S'-&c.;|-^}={|;[=&c.{^}={|,}, (for these quan. ties I -r- I &c. indicate the cosines of tlie angles which the directions of gravity makes with the three coordinates,) the three equations (n) may be made to assume the following form : they are satisfied by means of the three following: 0=2.mx; 0=^S.my ; 0=^2.mz. The equations (m) will be reduced to the following 92 CELESTIAL MECHANICS, The origin of the coordinates, being supposed fixed, it will destroy parallel to each of the three axes, the forces by composing these three forces, we shall obtain an unique force, equal to S.T.m. i. e. to the weight of the system. This origin of the coordinates about which we suppose the system in equilibrio, is a very remarkable point in it, on this account, that being supported, the system actuated by the sole force of gravity remains in equilibrio, whatever position it may be made to assume about this point, which is from thence denominated the centre of gravity of the system. Its position may be determined by this property, that if we make any plane whatever pass through this point, the sum of the products of each body,* by its distance from this plane, is equal to nothing ; for this S 5 S these forces admit a resultant, see note to pace 89, and as -r— , .^ , -^ , are equal to the ex dy dz cosines of the angles which its direction makes with the axes of .r, of y, and of z, com- bining those three expressions, the resultant is evidently = to Sim ; consequently the force with which the fixed origin is pressed, in this case equals the weight of the bodies com- posing the systems. S.lm. answers to the expression tng. in the first note to page 78. It follows from note to page 88, that the resultant of all the forces must pass through the ori^n for 2,rax ; l.my ; 2,n?2 ; are equal respectively to nothing. If another point in the system \)e$ides the centre of gravity was fixed, then 0 = S.\ — V i.my r— . l.mz. > is the sole equation of equilibrium ; in this case the fixed axis of rotation must be vertical. * If Ax' -{■ By'-\-Cz'= 0, be the equation of a plane passing through the centre of gra- vity, the cosines of the angles which this plane makes with the plane of the axes x y, of X z, and of y z, respectively, i. e. the cosines of the angles which a perpendicular to this plane makes with th,e axis of .r, and of y, of 2 = ABC see LacroLx, tom. 1. No. 269, in like manner the cosines of the angles, which lines drawn, from the point, whose coor- dinates are x, y, z, make with the axes of x, of y, and of z, • PART I.— BOOK I. 93 distance is a linear function of the coordinates x, y, z,, of the body ; consequently by multiplying it by the mass of the body, the sum of these products will be equal to nothing in consequence of the equa- tions. CoJ In order to determine the position of the centre of gravity, let X, Y, Z, represent its three coordinates with respect to a given origin j let x, y, z, be the coordinates of m with respect to the same point ; 'Z', y', z', those of m', &c. the equations (oj will then give O = x.m.(r — X.) but we have s.w.X=Xz.w, ■z.m being being the entire mass of the system, therefore we have y 7:.m.x we shall have in like manner s.wi.j/ ^.m.z j ^ — s.w ' -z.m ' :• by note to No. 2, page 7, the cosine of the angle which the perpendicular to the given plane, makes with the line whose coordinates are x, y, z, xA+yB +zC xA+yB+zC let this angle = a and ^/x=-f^^+s» x cos. a'= ./^a i P2 , /-z = ^^^ distance ot the point from the given plane, consequently, the sum of all the distances multiplied res- pectively into their masses A. ^.mx-\-B. J.my+ C. S.wz in consequence of the equation (o). 94 CELESTIAL MECHANICS, thus, as the coordinates X, Y, Z, determine only one point, it follows that the centre of a system of bodies is an unique point. The three preceding equations give this equation may oe made to assume the following form :* the finite integral s»zwj'[(y — or)* + (?/' — y)*+(:' — s)'] expresses the sum of all the products similar to that, which is contained under tlie charac- teristic s, and which is formed by considering all the combinations of * The square of the sum of any number of quantities, being equal to the sum of the squares of tho.-e quantities, and twice the sum of tUt; products of all the binary combi- nations of the different quantities, we iiave (2(m.r)) ' = 2()n^.T^ ) -|-2 ^[mm'.xx) ; Smm/. (x — x')') denotes the products which are obtained, by taking on one part all the binary combinations of the bodies mm , &c. in which the quantities mm are affected with different accents, and then multiplying these by the square of (x — x), in which the terms have respectively the same accents as tlie bodies which they are multiplied by, thus 2.(x — x')' = r^4-x'^-)-x*^ + &c. — 2xx' — 2xx" — 2x'x" — &cand ^{mm' x — .t'y) = mm'x'+mm'j/^ -\-mm".x' 4- nim"^"^ + m'm"x'^-j-ni'ni'..i"'-^-S:c. — Imm'xx — Imn'xx" — Im'm" xfx" ; &c. zz'S^mm'.x') — 22( )H///'.(xx')) and as 2(mx*) =, mx* +jn'x'^ +m"x"^ +&c. 2(ot^- ). 2m. =(jnx^+w't'*-; m''2x"» I &c.)-(»n+w'+''-f m"'-t-&c.) ■=:m-x^ + m"'x"--\-m"^x"' +&c, + mmx- -f-mHj'x'+jwm'x'' +m"7n'.c'^-^7nm".x''' ■f-m'm'.x"^ -\-&c. =2(m*x')4. 2(wm'x2) •.• (_;„.i) = = 2(wn = )+ 22(mm'xa') = 2(mx*). 2w — 2(»nm'x') — 2.ram'.(x— j;*)' -j-2;wm')x^) == 2(inx-).2m — 2OTm'(x — x )-, (by substituting lor 2(«i»r^) its value 2()nx")2m — 2'wm'.(x^;, and for 25(?wm'.xx'). its value 2(mni'.(x2),) — 2(^mm'.{x — x')\) •-• the value of Jt* (Smx)' (2»nx') 'Smm'^x — x)' (2jnj* "" im (2?h)« ' we might derive corresponding expressions for Y^, and Z*. This method gives the position of the centre of gravity of any body of a given fonn> •without being obliged, to refer the position of its molecules to coordinate planes. PART I.— BOOK I. 95 the different bodies of the system. We shall thus obtain the distance of the centre of gravity from any fixed point, by means of the dis- tances of the bodies of the system, from the same fixed axis, and of their mutual distances. By determining in this manner the distance of the centre of gravity from any three fixed points, we shall have its position in space ; which suggests a new way of determining this point. The denomination of centre of gravity has been extended to that point, of any system of bodies, either with or without weight, which is determined by the three coordinates X, Y, Z. 16. Is is easy to apply the preceding results to the equilibrium of a solid body of any figure, by conceiving it made up of an indefinite number of points, firmly united together. Therefore let dm be one of these points, or an indefinitely small molecule of the body, and let X, y, z, be the rectangular coordinates of this molecule ; also let P, Q, R, represent the forces by which it is actuated parallel to the axis of r, of J/, and of z, the equations (w?) and (ri) of the preceding number will be changed into the following : 0 =fP.dm ', O =fQ.dm ; 0 ^fR.dm-* 0 ^/C-Pi*— Q-^)- dm ; 0 =f{Pz—Rx). dm ; O =f{Ry—Qz). dm ; The sign of integration f is relative to the molecule dm, and ought to be extended to tlie entire mass of the solid. * 5 "V~ f being the cosine of the angle which the direction of the force S makes with the axis of x, S. ) — > = the force resolved parallel to the axis of x, ••• it is equal to P; and as^.m=/dm, 2m. -S.^-r^f =/ P.dm, and since 2.S. ^-^i.i/m=fPi/.dm; OX t 0X ^.m.S. { y- { ^ } - ^- { -^ } =/ {P})- Q^) dm, &c. From the values which have been given in the text for the coordinates of the centre of gravity, it is manifest that the position of this centre remains unaltered, whatever change may take place in the absolute force of gravity, •.• when bodies are transferred from one latitude to another on the surface of the earth, though the absolute weight varies, still the position of the centre of gravity is fixed. 96 CELESTIAL MECHANICS, If the body could only turn about the origin of the coordinates, the three last equations will be sufficient for its equilibrium.* * When any system of homogeneous bodies is in equilibrio, the centre of gravity is then the highest or lowest possible ; this is immediately evident from the principle of virtual velocities, for let the weights of any number of bodies m, ?h', m", be denoted by S, S', S"» &c. and let y, s, s", &c. represent lines demitted from the centres of the several bodies m, m', m", &c. on the horizontal plane; now if the position of the system be disturbed in an indefinitely small degree, we shall have, when the bodies of the system are in equili- brio, the equation of virtual velocities Sh+ S'.h'+S.h"+&c. = 0, consequently the quantity of which this expression is the variation, i. e. Si + S'«'-f-S'V'-t. &c. (= the entire weight of all the bodies composing the system, multiplied by the distance of the centre of gravity of the system from the horizontal plane, = s^.S.'^m.) is a maximum or mioimum, and as the weight of all the bodies of the system is always given, the distance of the centre of gravity of the system from the horizontal plane must be either a maximum or a minimum when the system is in equilibrio ; this being established, it is interesting to know the equation of the curve, in which the centre of gravity is lower than in any other curve whose points of suspension and length are given; the'investigation of this curve, which is termed the catenary, is very easy, it occurs in all the elementary treatises, the differential equationis of the following form [i^-\-g).dx~g. cos. c.vdx' -\-y'. It might be proved conversely, that when the distance of the centre of gravity from an horizontal plane is the greatest or least possible, the system is in equilibrio, for we shall have SJ.<-(-i)'.Js' + &/''3.^''/-^-&c. = G.S.« , =0, however there is an essential difference be- tween their states of equilibrium ; in the first case, the equilibrio is denominated instable, in the second, it is termed stable, in order to determine these two different states, we should attend to the species of the motion when the centre deviates by an indefinitely small quan- tity from the vertical, see Xo. 30. * In Physical and Astronomical problems, the method that is generally employed, to determine the mean value between several observed ones, of wliich some are greater, and some less than the true one, is to divide the sum of all the observed values by their number. Tliis comes, in fact, to determine the distance of the centre of gravity from a given plane. For if z, z, z", &c. represent the observed quantities, then , &c. n is the expression for the mean value, but if ^, 2', 7!', denote the distances of the centres of zm-\-z'mi-\-z"m."-\- gravity of n masses, equa each to m from the plane, then — , arc. = PART I.— BOOK I. 97 the distance of the centre of gravity of the system of in masses from this plane — '" , &c. = the required mean value. If several forces concurring in a point constitute an equilibrium, then supposing that, at the extremities of lines, in the directions of these forces, and respectively proportional to them, we place the centres of gravity of bodies equal to each other, the common centre of gravity of these masses will' be the point where all the forces concur. For since the forces are by hypothesis represented by lines taken their direction, and con- curring in one point, it is evident that by making this point tlie origin of the coordi- nates, we shall have the sum of the forces parallel to the three rectangular axes propor- tional to 2(j), 2(!/), 2(3), these sums are ••• by the conditions of the problem = to nothing, see note to page 11 ; and since the masses are all equal we shall have 2(x).m = 2(»a) = 0, this also obtains for the other axes, consequently we shall have 2(ot.j-) = 0, ~{m^) = 0, 'Z{mz) = 0, ••• the origin of the coordinates coincides with the centre of gravity of the system of masses respectively equal to vi. The centre of gravity of a body, oc system of bodies, is that point in space from which if lines be drawn to the molecules of the body, the sum of their squares is the least pos- sible. For if X, Y, Z, represent the coordinates of such a point, then the sura of the squares of the distances of all the molecules of the system from this point is equal to 2((j; — X) *-f-(y — ^*(z — Z)'^)> 'f ^e take the differential of tliis expression with respect to each of the coordinates, and multiply each of the terms of the sums which are respec- tively equal to nothing, by the element of the mass, we shall have 2.m.(.i- — .Y) = 0, 2.»7i.(^— Y)=0, 2.«4z— Z,) = 0, ... A'=r ^Oa^. Y_ ^^^ ■ Z— ""'" - 2w 2»J ' 2ffi and from what has been demonstrated in the preceding note it follows, that if we apply to all the points of the system, forces directed towards the centre of gravity, and propor- tional to the distances between those points and the centre of gravity, these forces will constitute an equilibrium ; consequently when several forces constitute an equilibrium, the sum of the squares of the distances of the point of concoiu-se of these forces, from the extremities of lines representing these forces, i. e. the sum of the squares of these Unes, is a minimum. From the preceding property it appears, that if several observations give different values for the position of a point in space, the mean position, i. e, the position which deviates the least from the observed positions, is that in which the sum of the squares of its dis- tances from the observed positions is the least possible. The problem is altogether similar when we wish to combine several observations of a7iij kind whatever; for the distances of the points correspond to the differences between the particular results and their mean value ; and since it is impossible entirely to exterminate these differences, we are obliged to select a mean result, such that the sums of the squares of these differences may be a mi- O 98 CELESTIAL MECHANICS, niniuni ; this is the principal of the method of the least squares, which was devised by Le Gendre to combine the equations of conditions between the errors deduced from a. comparison of the astronomical tables with observation ; it comes in fact to find the centre of gravity of the observations which we compare together. The general form of the equations of condition is as follows : 0 ~ a-\-bx-\-ci/-\-dz-\-&c. when we pass into one member all the terms which com- pose them, a, b, c, are given numerical coefficients, if all these equations could be satis- fied exactly, by the values of x, y, z, their first members would be necessarily reduced to nothing by substituting for x, y, z, their values, but as this substitution does not render them accurately equal to nothing, let E, E', E", represent the errors which remain, then we shall liave E— a-\-bx-\-cy-\-dz-^&:z. ; E~ a'-hi'x-f c'y + i/'s-f&c. ; E":=.a" \V'x-\-c'iy^ &c. the quantities x, y, z\ &c. are to be determined by tlie condition that the values E, E', are either nothing, or very small ; the sum of the squares of the errors := i;=-fii=+£'-4-&c.= (a=-fa'--fa'''=4-&c.)-i-{i'+i'*+i''''*).x2+(c--f (■' = -ff»'*-f&c.)/- -f.(rf*-ha!"- -\-dff ' -f &c.)« ' -1- ; 2{abJro:b'^a"b"->r&c.)xJ(- 2 {acJra'c'+a"c«)y{- 2 {ad-\-dd'+a''d''-\-8cc.)z; ■\-%bc+b'c' -\-b/'c"J^&c.) xy-{- ^(bd + b'd'-^b"d'/)-{-^z + &iC. the minimum of this expression, with respect to x, will be 0 = 2.n6+x 'S-.h'^+y l..bc^z. 2.M-f &c. the minimum with respect to j/ = 2.ac-|-x2.6c-|-y2c*-l- x.2.rfc= 0, we derive a cor- responding value for the minimum of z, hence in order to form the equation of the minimum with respect to one of the unknown quantities, we must multiply all the terms of each proposed equation by the coefficient of the unknown term in that equa- tion, and then put the sum of the products equal to nothing. Though this method requires more numerical calculations, in order to form the particular equation relative to each unknown quantity, than the method suggested by Mayer ; it is more direct in its application, and requires no tentation on the resulting equations. Laplace has shenii in his Theory of Probabilities, that when we would take the mean between a great number of observations of the same quantity, obtained by different means, this is the only method which the theory permits us to employ, see Le Gendres Memoir on the determination of the orbits of the comets, ard Biot's Astronomic Physique, tome 2. page 200. PART I.— BOOK I. 99 CHAPTER IV. Of the equilibrium of fluids. 17» In order to determine the laws of the equilibrium, and of the motion of each of the molecules of a fluid, it would be necessary to ascertain their figure, which is impossible ; but we have no occasion to determine these laws, except for fluids* considered in a mass, and for this purpose the knowledge of the figures of their molecules is useless. Whatever may be the nature of these figures, and the properties which depend on them in the integrant molecules, all fluids, considered in the aggregate, ought to exhibit the same phenomena in their equilibrium, and also in their motions, so that from the observation of these pheno- mena, we are not able to discover any thing respecting the configura- tion of the fluid molecules. These general phenomena depend on o2 * Although the figure of the molecules of fluids are unknown to us, still there can be no question but that they are material, and consequently that the general laws of the equili- brium and motion of solid bodies are applicable to them. If we were able analytically to express their characteristic property, to wit, extreme smallness, and perfect mobility, no particular theory would be required in order to determine tiie laws of their equilibrium and motion ; they would be then only a particular case of the general laws of Statics and Dy- namics. But as we are not able to effect this, it is proposed to derive the theory of their equilibrium and motion from the property which is peculiar to them, of transmitting equally, and in every direction, the pressure to which their surface is subjected; this property is a necessary consequence of the perfect mobility of the molecules of the fluids. In the definition wliich has been given in the text there is no account made of the tena- city or adhesion of the molecules, wloich is an obstacle to this free separation ; this adhe- sion exists however between the molecules of most of the fluids with which we are ac- quainted. 100 CELESTIAL MECHANICS. the perpect mobility of these molecules, which are thus able to yield to the slightest force This mobility is the characteristic property of fluids ; it distinauishes them from solid bodies, and serves to de- fine them. It follows from this, that when a fluid mass is' in equilibrio each molecule must be in equilibrio in consequence of the forces which * solicit it, and of the pressures to which it is subjected by the action of the surrounding particles. Let us proceed to develope the equations whicii may be deduced from this property. For this purpose, let us consider a system of fluid molecules, consti- tuting an indefinitely small rectangular parallelepiped. Let x, y, z, denote the three rectangular coordinates of that angle of the parallele- piped, which is nearest to the origin of the coordinates. Let dx, dy, dz, represent the three dimensions of this parallelepiped ; let p repre- * When a fluid is contained in a vessel, the pressure to which it is subjected at its sur- t'ace is transmitted in every direction, as has been just stated, but since the molecules ar^ material, 'they must have weight, therefore it also presses the sides of the vessel with a force arising from the weight of the molecules, and different in every point of the sides ; and if the fluid is contained in a vessel closed in every side, when the molecules are solicited by any given accelerating forces, then the pressure is different for every particular point, its direction is always perpendicular to the surface, since by No. 3, when the resistance of a surface destroys the pressure on it, the direction of this pressure must be normal to the sur- face. The intensity of this pressure depends on the given forces, and on the position of the point. Therefore it appears, that in the equilibrium of a fluid contained in a vessel, the entire pressure in each point of the sides is the sura of two pressures altogether distinct ; one of which arises from the pressure, exerted on the surface, and is the same on all the pomts ; the other is owing to the motive forces of the particles of the fluids, and varies from one point to another. Fluids are generally distinguished into two classes, incompressible, and elastic; with respect to the last class, they may press against the sides of the vessel in which they are enclosed, although no motive forces act on the particles, or without any pressure urging the surface of the fluid. For from their elasticity they tend perpetually to dilate them- selves, which gives rise to a pressure on the sides of the vessel : however this is a constant pressure in tlie same fluid ; it depends on the matter of ll>e fluid, its density and tem- perature. PART I— BOOK I. 101 seut the mean of all the pressures, to which the different points of the side dy. dz of the parallelepiped, which is nearest to the origin of the coordinates, is subjected ; and let f' be the corresponding quantity on the opposite side. The parallelepiped, in consequence of the pressure to which it is subjected, will be urged in the direction of x\ by a force equal to (j) — ;p')- dy.dz ; p — p is the difference of f, taken on the hy- pothesis that .r alone is variable ; for although the pressure f acts in a direction contrary to ^j, nevertheless the pressure to which a point is subject being the same in every direction, p'*-p may be considered as the difference of two forces infinitely near, and acting in the same di- rection ; consequently we have* p'^^zz < -J-^\.dx, and [p — p'). dy. dz = — \~J^ \' ^'^' ^y' ^~' Let P, Q, /»*, be the three accelerating forces which solicit the mo- lecules of the fluid, independently of their connexion, parallel to the axe^ of X, of y, and oi z ; if the density of the parallelepiped be denoted by />, its mass will be equal to p. dx. dy. dz. and the product of the force P by this mass, will represent the whole motive force, which is derived from • Since p, 5, P,Q,Il, generally vary from one point to another of the fluid mass, tliey must be considered as functions of x, 7/, z. We distribute the fluid into parallelepipeds, in order more easily to express in analytical language the fact of the equality of pressure, which, as has been stated, is the fundamental principle from which we deduce the whole theory of thtir equilibrium, and by supposing these parallelepipeds indefinitely small, we ite permitted to consider all the points of the same side as equally pressed, and also ^ > Pt Q, R, as constant for each side respectively, by means of which we are able to detennine the pressure p. x, y, z, being the coordinates of the angular point next the origin, and p being a function of these coordinates, we shall have the coefficient ) ~- [ = < -^ \ &c. they are taken negatively because they tend to diijamish the coordinates. Sz JOS CELESTIAL MECHANICS, it ; consequently this mass will be solicited parallel to the axes of x\ by the force jpP— f— f-1 C, dx.dy.dz. For similar reasons it will be so- (. \ dx SS licited parallel to the axes of j/, and of a, by the forces ip.Q — \-~- 5 f . dx.dy.dz. and S^.R — 1-^|^. dx. dy. dz. &c. therefore, by the equation [b) of No. 3, we shall have or ip = p(P.<J*4-Q.J;j/+ii.J^). The first member of this equation being an exact variation, the second must be so likewise ; from which we may deduce the following equation of partial differentials,* I dy S~ I dx )' I dz ^ \ dx f I dz >~C dy i . * Wheng (P.Jx+Q.Sy+/?.Ss.) is an exact difFerential, | — ^| — 1= < -~ \ &c. (see Lacroix Traite Elementaire, Calcul. Differential and Integral, No. 261.) '''~~clf"^ Ihj dx "^ dx ' dz "^ dz ~ dx "^ dx ' SL—L . _^?:iL__ ± 1 i_Lj if we multiply the first equation by R, the second by dz ~ dz dy dy — Q, and the third by P, we shall obtain, ^.R.dP: JhP.d^ R-i-dQ R.Q.d^ ^.Q.dP Q.P.di _ ^.Q.dR dy ^ dy ' dx dx ' dz dz dx R.Q.d^ ^.P.dQ P.QJj _ ^.P.dR I R.P.di dx ' dz dz '^ dy dy ' PART I.— BOOK I. 103 from which we may obtain This equation expresses the relation which must exist between the forces P, Q, and R, in order that the equilibrium may be possible. If the fluid be free at its surface, or in certain parts of this surface, the value of p will be equal to nothing in those parts 5 therefore we shall have Sp z: 0, provided that the variations Sx, Sy, iz, appertain to this surface ; consequently when these conditions are satisfied, we shall have O = PJx + QJy + RJz. If Su — O, be the differential equation of the surface, we shall have PJx + QJt/ -jr R.iz = \Ju, X being a function of r, 1/, z', from which it follows, by No. 3, that by reducing all the terms in which Jj is involved to one side, and then adding them toge- ther, we get ( RMP R.dQ Q.dP , Q.dR P.dQ P.dR y. dy dx dz dx dz dy _RP^ RQ.d^ QP.d^ RQ.d^ PQ.3g RP.d^ _ dy '^ dx ■*■ dz ~ dx ~~ dz + dy "^ ^ * by coacinnating This equation shews whether the equilibrium is possible, though we are unable to as- certain the density ;. 104 CELESTIAL MECHANICS, the resultant of the forces P,Q,R,* must be a perpendicular to those parts of the surface, in which the fluid is free. Let us suppose that the variation P(}'x 4- Q.(r3/ + i?.(?s; is exact, this is the case when P,Q,R, are the result of attractive forces. Denoting this variation by J'<?, we shall have Sp = fS(p; therefore p must be a function oi p and of 9, and as the integration of this differential equation gives (p If the relation indicated by this equation does not obtain between the forces P, Q, R, the fluid will be in a perpetual state of agitation, whatever figure it may be made to as- sume ; but when this relation is satisfied, the equilibrium will be possible, and vice versa ; and as P,Q,,R, are functions of the coordinates, we can integrate the expression 5.(P.Sx+ Q.Sy-[- J?.S;.) by the method of quadratures, by means of which we can find the value of the pressure for any given place of the fluid ; consequently we can obtain the force with which any side of the vessel in which the fluid is enclosed is pressed. But though the relation which exists between the forces must be such as to satisfy the preceding equa- tion, when there is an equilibrium, still this is not suflicient, in most cases, to insure the equilibrium, for the fluid must also assume a determined figure, depending on the nature of the forces P, Q, R, which solicit the molecules. * When an imcompressible fluid is free at its surface, and in a state of equilibrium, p must vanish, v Sp—0, if the fluid is elastic this condition can never be satisfied, because g being proportional to p, whilst the density has a finite value, p can never vanish. When p vanishes, Or:J/^=P.Sx-{- Q.Sy-f ii.Sx, v when Sx, S^, 3;:, appertain to the surface, by sub- stituting for P, Q, R, their values, the resulting expression will be the equation of the sur- face. It follows from No. 3, that the resultant of the forces P, Q, R, must be perpendi- cular to the surface ; it may be proved directly thus : P Q R y/p^^a'-VR" V/'^ + Q-fA'*' V P' + Q.''+R^' are equal to the cosines of the angles, which the resultant makes with the axes of x, ofy, and of z, but since P.Sx-j-Q.Jy+iJ.Jz, is the equation of the surface, they also express the cosines of the angles which the normal make with the same axes respectively ; see Notes to page li ; consequently the normal coincides with the resultant. This coincidence of the resultant with the normal is the second condition, which must be satisfied, in order, as has been stated above, to insure the equilibrium ; and it is this condition which enables us in each particular case to determine the figure corresponding to the equihbrium of the fluid, and if there be one only attractive force directed tov.ards a fixed point, then the surface will be of a spherical form, the fixed point being the centre of the sphere ; if this point PART L— BOOK I. 105 in a function ofp, we shall have p in a function of />. Therefore the pressure is the same, for all molecules whose density is the same ; thus Jf> must vanish with respect to those strata of the fluid, in which the density is constant, and with regard to these surfaces, we have, 0=PJx+QJy+RJz.* consequently, the resultant of the forces, which solicit each molecule P be at an infinite distance the surface will degenerate into a plane, ••• if the planets were 1 originally fluid, and if their molecules attracted each other with forces, varying as -— they would assume a spherical form. See No. 12, Book S**. • If P.3x-f-(J.Jy+/?.32is an exact variation, Sip, Jprr^Jip, •.• j must be some function of <p, otherwise it would not be an exact variation ; however, the form of this function is undetermined, see note to page 10, consequently p will be a function of ^, and p and { will be the same for all those molecules in which the value of (p is given, i, e. for the molecules in the same strata of level, therefore when the density varies, an equili- brium cannot subsist unless each stratum is homogeneous during its entire extent ; for when this is the case, j, and consequently p is the same ; ••• ^p :x. 0, for the surfaces in which 5 is constant, •.• for such surfaces O^P.Jj^+ Q.5y-j-i2.Jz, and the resultant coin- cides with the normal. If we integrate the preceding equation, by putting ip equal to a constant arbitrary quantity, we derive an equation which appertains to an indefinite num ■ ber of surfaces, differing from each only by the value of this constant arbitrary quantity. If we make this quantity increase by insensible gradations, we will have an infinite series of surfaces, distributing the entire mass into an indefinite number of strata, and constituting between any two successive surfaces, what have been denominated strata of Level. The law of the variation of the density 5, in the transit from one strata to another, is altogether ar- bitrary, as it depends on what function of ip, 5 is, but this is undetermined. It appears from what precedes, that there are two cases, in which Sp — 0, when it is at the free surface, in which case p must vanish of itself, and also when p is constant, i. e. for all surfaces of the same level, consequently when the fluid is homogeneous, the strata to which tlie resultant of the forces is perpendicular, are then necessarily of the same density. When the fluid is contained in a vessel, closed in on every side, it is only necessary that all strata of the same level must have the same density ; in elastic fluids, the first condition to wit, that p should vanish, or that P.3j;-|-Q.Sy-|-i?.Jz— 0, can never obtain, v unless this fluid extends indefinitely into space, so that { may be altogether insensible it cannot be in equilibrium, except in a vessel closed in on every side. 106 CELESTIAL MECHANICS, of the fluid, is in the state of equilibrium, perpendicular to the spruces of these strata, and on this account they have been termed strata of level. This condition is always satisfied, if the fluid is homogeneous, and incompressible, because then the strata, to which this resultant is perpendicular, are all of the same density. For the equilibrium of an homogeneous fluid mass, of which the ex- treme surface is free, and covers a fixed solid nucleus of any figure whatever, it is necessary and sufficient, first, that the variation P5x+ Q..Sy + R-\-Jz be exact ; secondly, that the resultant of the forces at the exterior surface be directed perpendicularly towards this surface.* * If two different fluids are in equilibrio, then the surface which separates them must be horizontal ; if the denser fluid is superior, the centre of gravity of all the molecules will be highest; if it be inferior, then the centre will be lower than in any other position, •.■ that the equilibrium may be stable, the denser strata should be inferior. See Kotes to No. 15. When ^ is constant, the equation ip^C, gives the relation which must exist for each stratum of level between the coordinates of the different molecules of the surface which answers to the preceding equation ; in this case S<p = 0, which shews that <p is either a maximum or minimum, and generally when P.Jc-j-Q.Jy+if.Ji is an exact variation, 5 is a function of $, •.• the equation of equilibrium 2yj — ^.S^p^rO, shews that in the state of equilibrium there is a function of p and of x, y, 2, which is either a maximum or a mini- mum. Though in the state of equilibrium all the molecules in the same strata of level have necessarily the same density, and experience the same pressure, still the converse is not true, for in homogeneous incompressible fluids, g is constant in those sections of the fluid in which neither 2ip, nor 5p~0. In elastic fluids, the density g is observed to be proportional to the compressing force, •.• p ■^.k.^ ; k depends on the temperature and matter of the fluid, by substituting for 5, in the equation J/j^zgSip, weobtain ep=.^. S<p, •■• by integrating we get log.p+C zz—, because when the matter and temperature are given, k will be constant, •••by makuig C= — log. £, <P we obtain piz Ec k, :• since p and 5, == ^ -j- j- , are respectively functions of <f), the pressure and density will be constant for each stratum of level, but the law of the variation of the density is not arbitrary, as in the case of incompressible fluids, for the, eqiiatiop p] E ^ .'' '■■'■""•■ 5 = ■—=: -J- . c k, determines the law. If the matter of the fluid remaining homo- geneous, the temperatuie undergoes any alteration, k will be a function of the variable PART I.^BOOK I. 107 tetnperatuire, but in order that the equation — ^s=— may be an exact variation, it is necessary that k, and ••• the temperature should be functions of <f), these functions are altogether arbitrary ; consequently we conclude, that when the fluid is in a state of equi- librium, the temperature of each stratum is uniform, and that the law of the variation of temperature is arbitrary ; but this law being given, we are able to integrate the expres- sion -y-i from which integral we can conclude the law of the densities and pressures by means of the equations p: . - ' ; — In incompressible fluids, if the force varies as the n"" power of the distance from the centre, by fixing the origin of tlie coordinates at this point, we have P ^ ^jr"-*. j, Q =: ^jr"-l y, R=A^r^Kz,-- P.Jx+Q.Sy-f-iJ.Sz =^g,'^l. (x.SxH-^%+J.3z)=Jjr". Jr, = dp, ••• , , when » IS given, r: (p =p+C, when n = — 2, —2 = n = ^ 1+1 n+l r if gravity is the sole force acting on the molecules, by making the axis of z vertical, P and Q, will vanish, and R = g, '.• /P.Sx-j- Q.dy + R.h is reduced to the equation g.h = 0, •.• gz = C, consequently the surface is horizontal, since R = (A^r'^^.z) =g,/(g.dz) =p :• the pressure varies as the height. Since when the force varies as the n"" power of the distance from the centre dp = Ar".dr^, by substituting in the equation of elastic fluids in ip Ar^"^^ ~~~T ft"" »^) and integrating, we get log. p = — >, consequently, if the P " K.(?2-j-l ' ("+ l)"" powers of the distance be taken in arithmetic progression, the pressures and the densities proportional to them, will be in geometric progression, •■• if n is negative, and if in the radius, ordiaates be erected proportional to the pressures or densities ,the locus of their extremities will be a curve of the hyperbolic species, and the radius produced, will be an asymptote to the curve, if n is positive, the locus of the extremities of the coordinates, will be a curve of the parabolic species, if n:;0, i. e. if the force is constant, the locus will be the loganthmic curve. See Princip. Matth. Liber 2. Prop. 22, et Scholium. p2 108 CELESTIAL MECHANICS, CHAPTER V. The general principles of the motions of a system of bodies. 18. We have, in No. 7,* reduced the laws of the motion of a point, to those of its equilibrium, by resolving the instantaneous motion into two others, of which one remains, while the other is destroyed by the action of the forces which solicit the point ; we have derived the diffe- rential equations of its motion, from the equilibrium which subsists be- tween these forces, and the motion lost by the body. We now proceed to employ the same method, in order to determine the motion of a system of bodies m, m', m \ &c. Thus, let mP, mQ, Em, be the forces which solicit ?« parallel to the axes of the rectangular coordinates *, y, z ; let m'P', m'Q', m'R', be the forces which solicit m, parallel to the same axes, and so on of the rest ; and let us denote the time by /. The partial forces m.—-—,m.-^,wz.—^ of the body m at any instant at at at whatever will become in the following :t • The principle established in this number, has been termed <Af pmiciple of D' Alembert, by it the laws of the motion of a system are reducible to one sole principle, in the same manner as the laws of the equilibrium of bodies have been reduced to the equation {I) of No. 14. \ In consequence of the mutual connection which subsists between the different bodies of the system, the effect, which the forces immediately applied to the respective bodies would produce, is somewhat modified, so that their velocities, and the directions of their motions, are different from what would take place, if the bodies composing the system were altogether free ; consequently, if at any point of time we compute the motions which / PART I— BOOK I. 109 m.—r + m. d. — m. d.—--+ mP.dt ; dt dt dt »i.-4^+ m,d. -4- dt dt m.d. ^ + mQ.dt dt dz , . dz m.—— + m. d. — — dt ' dt — m. d — r^+ inR.dt ; dt the bodies would have at the subsequent instant, if they were not subjected to their mu- tual action ; and if we also compute the motions, which they have in the subsequent in- stant, in consequence of their mutual action, the motions which must be compounded with the first of these, in order to produce the second, are such as if they acted on the system alone, would constitute an equilibrium between the bodies of the system ; for if not, the second of the abovementioned motions would not be those which actually ob- tain, contrary to the hypothesis. But as these motions, which must be compounded with the motions which actually have place, in order to produce the first, are altogether un- known ; in the analytical expressions, we substitute expressions equivalent to them, i. e. the quantities of motion which have actually place, taken in a direction contrary to their true one, and the motions which would take place, taken in the true direction, by means of this we are able to establish immediately equations of equilibrium between the iirst and second of the abovementioned species of motion, and also to determine the veloc.fies which would take place, if the bodies composing the system were altogether free. Now if we suppose the preceding motions, resolved respectively into three others parallel to three rectangular coordinates, mP, niQ, inR, m'P', &c. will represent the motions parallel to the three axes which the bodies would assume, if they were altogether free. d^x d'u d^z , d-^x m.- , _ , m. — r^ , m. — r--, m — , &c. dt^ dt' dt^ dt represent the motions parallel to the same aXes, which the bodies actually have, at the com- mencement of the secondinstant. Since the motions which actually take place, are to be taken in a direction contrary to their true one, they are affected with negative signs. We might by means of this principle, without introducing the consideration of virtual velocities, derive several important consequences ; but it is the combination of this prin- ciple with that of virtual velocities, which has contributed so much to the perfection of Mechanics ; this combination was first suggested by L'Agrange, who by this means has reduced the investigation of the motion of any system of bodies, to the integration of differential equations ; thus we can reduce into an equation every problem relating to Dy- namics, and it belongs to pure analysis to complete the solution ; so that it appears that the only bar to the complete solution of every problem of Mechanics, arises from the im- perfection of the analysis. It is manifest from the introduction of the expression —r-^> in P^^'® "^ ^^'^ increase 110 CELESTIAL MECHANICS, and as the forces rfj , dx dy , , dy dz ^ , dz ni. 1- m. d. —— ; m. -~+ in. d. -^- ; m. —— + m. d. — — ; dt dt dt dl dt dl of the velocity, tliat the changes in the motions of tlie body arc made by insensible degrees. The inspection of the equation (/') sliews tliat it consists of two parts entirely dis- tinct, of which one is the quantity which wc ought to put equal to nothing, when the forces P, (I, It, J'', itc. which are applied to the diflbrent points of the system, constitute an equilibrium, the other part arises from the motion which is produced by the forces /', <i, II, /■•', iVc. when they do not constitute an equilibrium ; therefore we may express the equation (/') in this manner: 0 = 2.(«.(/^.3x+a3y-f./?.3.-)- -m.^^.ix + ^. 33,+.-^'| . 3,. J ai\d the equation (/) of No. H', is only a particular case of the equation (F) ; thus the principle of virtual velocities may be considered as an universal instrument which is ne- cessary for the solution of all problems relating to Mechanics. The expression by which the equation (P) differs from the equation (l) is entirely independant of the positiim of the axes of the coordinates ; for by substituting the coordinates .r', t/', z, in place of the preceding coordinates x, y, z, by the known formula: we have X r: ax'-\-bff'+cz', , 1/ r= ax'Jfb'y'-^-c'z, the origin being tlie same, by differentiating the preceding expression twice, the coeffi- cients a, b, r, a, &c. being constant, we obtain n'»i = a.d^x'-\-b.d^iy -^-c.d^z', d*y = a'.d'x'+b\d't/+c.d*z-, d^z—of.d'x'+b^.d'y'+OM'x': PART I.— BOOK I. Ill only remain j the forces — m. d. + P. dt -.-^m.d. —^ + Q.dt ; — m.d. h R.dL dt ' dt dt will be destroyed. By distinguishing, in this expression, the characters wi, x, y, z., P, Q, R, by one, two, marks, &c. successively, we shall have an ex- pression for the forces destroyed in the bodies rd, m'., &c. This being premised, if \ve multiply these forces by the respective variations of their directions ix, Sy, Sz, &c. we shall obtain, by means of the princi- ple of virtual velocities, laid down in No. 14, the following equation, in which dt is supposed to be constant. I dt* S ( dt' ) ( dt' ■,(P) From this equation we may eliminate, by means of the particular conditions of the system, as many variations as we have conditions ; and then by making the coefficients of the remaining variations separately and also, 3x = a.Ji'-f-A.Jy'-J-c^z', h/^ a'.3f'-|-i'.3y+c'Ji', 3z = a' .3x'+ b^'.hj" + c".'i:f ; V by substituting for these expressions in the expression rf»x . , dry , d\z d*jf .,, dW ,,, d'^z ,, „. _. 3^+«._^.+^.__.weget,«.^^. Jx-hm.-^. b +'»-;^- ^^- {oT a^ -\-a'^ +a"' = 1, ab-^ac + 6c = 0, &'C. see Notes to page?; the same fiubstitu- tions being made in the expressions of the mutual distances between the bodies, the co- efficients a, b, c, a', &c. will disappear for the same reasons. 115 CELESTIAL MECHANICS, equal to nothing, we shall obtaui all the equations necessary for deter- mining the motions of the several bodies of the system. 19. The equation (p) involves several general principles of motion, which we shall examine in detail. The variations Sx, Sy, Sz, will be subjected to all the conditions of the connection of the * parts of the forces, by supposing them equal to the differentials dz, dy, dz, dx'. Sec. * If the equation of condition involves the time explicitly, then we are not permitted to suppose the variations Sx, hj, Sz, equal to the differentials dx, dy, dz, as for instance, if one of the bodies composing the system, always existed on a given surface, which surface moved according to a given law ; or if the body moved in a resisting medium, whicli medium was in motion, then there will exist an equation between the coordinates of the body and the time which will also be at any instant, the differential equation of the sur- face, the most general equation expressing the preceding condition, is of the following form: 0.{x, y, z ; x, y, z, &c. t) =0, at the following instant the coordinates will be varied by the quantities 3j, ly, Iz ; Sx', oy\ &c and the equation of condition will become <p.{x-\rSi, y ^Sy, zlzx x'+J^, y'+Sy', s'+Sz', &c. t) =:F =r. 0, V the difference of these two expressions, i. e. bat the complete differential of the preceding function = T is the differential coefficient of F, taken on the hypothesis that the time varies, conse- quently, if F involves the time explicitly, when we subject the variations Ix, iy, &c. to satisfy the conditions of the connection of the parts of the system, we are not permitted to regard the expression as equal to nothing. PART I— BOOK I. 113 This supposition is consequently permitted, and then the integration of the equation (P) gives V ,„.i^il±J^±f^) =c+2.2.>.(P.di + Q.^i/+i2.(/^) ; (Q) c being a constant arbitrary quantity introduced by the integration. If the forces P, Q, R, are the results of attractive forces, directed towards fixed centres, and of a mutual attraction between the bodies ; the function "L.fm.^P.dx + Q.dy+Rdz)* is an exact integral. For the Q * In fact, the accelerating force of m, produced by the action of m in the direction of the line_^ zsm F, (Fis always a given function ofjl) '.' the components of tliis force pa- rallel to the axes of x,i/, z, are m'F. - — - — , m'F. , m'F. ~ , •.' the parts of P.dx+ Q,.dy-\-R.dz, which answers to this force alone are ni'F.{{x' — x).dx-^{y' — y).dy-\-{z' — z).dz), and as the accelerating force of m', arising from the action of m, resolved parallel to the coordinates x, y, z, respectively = m.F. — 4- m.F. ^^~y + VI.F. ~~ , the corresponding part of FJx-t Q'.dy'+R'.dz, is, F.w. |if=ii.rf/+l^^:^.%'+if=il. dz\, therefore in or- der to have the motive force ai'ising from the mutual action of the bodies m and m' we must multiply the first expression by m, and the second by »n', and adding them toge- ther, they will become mm'.F. (x'—x). dx + (y'—y).dy+(z-zJ.dz+{x—x').dx^-\-{y—y'). dy+iz—z), d^)= mm'.F.fd/, for asf^ = {x-^y+{y-yY +{z-z'y,fdf:^ {x-x').(dx-dx') + {y-yWy-dy) + {z-z).{dz-dz), consequently as F is given to be a function of y; ^f-dj. is an exact differential. If the .centres to which the forces are directed Jiave a motion in space, then P.rfx+ Q.di/^Rdz, is not an exact differential, though the law according to which the forces vary should be a function of the distance, see Note to page 34. The sum of the living forces at any instant will be given by the equation ( Q), when we know the value of this sum at a determined instant, and the coordinates of the bodies composing the system in the two positions of the system. And when the system returns to the same position, the living forces will be the same as before. 114. CELESTIAL MECHANICS, part whicli depends on the attractions directed towards fixed points, are exact integrals by No. 8. This is equally the case, with respect to those parts, which depend on the mutual attractions of the bodies com- posing the system ; for if we name^ the distance of m from m', m'F, the attraction of ot' on »z ; the part of m(P.ds + Q.di/ + R.dz) which arises from the attraction of m' on m, will be, by the above cited No. equal to — mra'Fdf, the differential df being taken on the supposi- tion, that the coordinates x, y, z, only vary. But reaction being equal and contrary to action, the part o?m'{P'.dx'-\-Q'.dy'-]rR'dz'^ which is due to the attraction of m on m', is equal to — mm'.Fdf, the coordi- nates x', y\ z', being the only quantities which are supposed to vary, consequently df being the differential of y on the supposition that both the coordinates x, y, z, and x', y', z', vary simultaneously, the part of the function 1.77i(^P.dx-{-Q.dy -h Ji-dz) which depends on the reciprocal action of m on vi' is equal to —nim'.F.d/i Therefore this quantity is an exact differential when F is a, function of f, or when the attraction varies as some function of the distance, which we shall always suppose ; consequently the function 1.7n.(P.dx+Q.dy-{-R.dz} k an exact dif- ferential, as often as the forces which act on the different bodies of the system, are the result of their mutual attraction, or of attractive forces directed towards fixed points. Let then d(p represent this differential, and naming v the velocity of 7n, t/ the velocity of ni', &c. we shall have I..mv' = c + 2<?. (R) This equation corresponds to the equation (g) of No. S, it is the analytical expression of the principle of the conservation of living forces. The product of the mass of a body by the square of its velocity, is tenned the living force, or the vis viva of a body. The principle just announced consists in this, that the sum of the living forces, or the entire living force of the system is constant, if the system is not b,oIicited by any forces ; and if the bodies are actuated by any forces whatever, the sum of the increments of the entire living force is the same what- PART I.— BOOK I. 115 ever may be the nature of the curves described, provided that their points of departure and arrival be the same.* However this principle is only applicable, when the motions of the bodies change by imperceptible gradations.! If these motions undergo abrupt changes, the living force is diminished by a quantity which may be thus determined. The analysis which has conducted us to the equation (P) of the preceding number, gives us in this case, instead of that equation, the following : 0= S.w. ^—-.A. -— + -f-.A. 4-+ -—.A. ( dt dt dt dt dt dz IF q2 * What has been demonstrated respecting the mutual attraction of the bodies of the system, is equally true respecting repulsive forces which vary as some function ot' the distance; it is true also when, die repulsions are produced by the action of springs in- terposed between the bodies'; for the [force of the spring must vaiy as some function ot the distance between the points, •.• in the impact of perfectly elastic bodies though the quantity of motion communicated may be increased indefinitely, stUl the living force after the impact is the same as before ; indeed during the impact, the vis viva varies as the coordinates of the respective points vary, but after its completion, from the nature of perfectly elastic bodies they resume their original position, and consequently the value of the vis viva will be the same as before, but if the elasticity is not perfect, in order to have the value of the vis yiyaat an}^ instant, we should know the law of the elasticity, or the relation which exists between the compressive and restitutivc force. J WTien the motions of the botlies of the system, are modified by friction, or the re- sistance of the medium in which the motion is performed, the expression P.dx-\-Q,.dy-\-Rjdz is not an exact differential, see note to page 34, and the living forces must be diminished. This is indeed evident of itself, for when the bodies of the system are actuated by no other forces but those of resistance, the sum of the living forces must be gradually dimi- nished, in order to determine the actual loss experienced after any time, we should know the law according to which the resistance varies, which is very difficult to be determined ; but there is another cause of diminution of the living force, in which we are able to deter- mine accurately the loss sustained, to wit, the case adverted to in the text, when the bodies undergo an abrupt change in their motions. X The characteristic A designates according to the received notation, the difference which exists between- two consecutive states of the same quantity. 116 CELESTIAL MECHANICS, -^x.m.{PJx-i-QJi/^RJz) ', dx du , dz , . . ...rr, r dx dy dz r A. — , A. -^ , A. — — , being the ditterences of — j-t —r-> —r-, trom dt dt dz *' dt dt dt one instant to another ; differences which become finite, when the motions of the bodies undergo finite alterations in an instant. In this The equation (/-") may be made to assume the following form : in which tlie changes that are produced in the motions of the bodies composing the system, are made by insensible degrees, as is evident from the circumstance, that the differential of the velocities is expressed by — — , see note to page 30 ; now, if instead of this gradual di- 1 • • dx dy ininution, bodies experience abrupt changes m their motions A. -^. A.— -, &c. express- ing those changes, the preceding expression will be changed into the following : -2.n;.( P.Sj+ Q.?^ -J- jR- Jy ; dx Sx ~di" It dx and as in this case wA. — ^ is the variation of the force of the body, on the supposition dt that it is entirely free, and m.P.dt is the variation which actually takes place in conse- quence of the action of the bodies of the system, the reasoning in No. 18 is applicable to this case, consequently the preceding expression may be put equal to nothing ; and since the values of dx, dy, dz, are changed in the following instant into rfx+A.tfor, dy-\-^-dy. dz+^dz, we shall satisfy the conditions of the connection of the parts of the system, by making tlie variations 3x, iy, h, equal to these expressions respectively ; and then the preceding equation will assume this form f dx _ dx t dx , ( dy dy \ dy { dz_ Jz_\ dz^ ' dt"^ dt ] ^' dt 2.»n.(P. ( (ix+A.rfx ) + Q. ( rfy-f- A.rf^) + i?.((/z+ A.(/£ ), =0, PART I— BOOK I. 117 equation we may suppose Sj:=:dx+A.dx ; $y zz dy + A..dij; Sz = r/^+A. dz; because the values of dx, dy, dz, being changed in the following in- stant into d.T + £^.dx, dy + £:i,dy, dz+A>..dz, these values of Sx, Sy, Sz, satisfy the conditions the connection of the parts of the system ; there- fore we shall have \^ dt^ dt ^ dt^^dt^ dt ^ dt C dz dz > dz } i-dF^ ^•~dfy^~di\ ^ X.m.(P.(.dr+A.dx) + Q.(dy+A.dy+B.(dzi- A.dz)) This equation should be integrated as an equation of finite differences relative to the time t, of which the variations are infinitely small, as well as the variations of ^, y, z, a/, &c. Let 2, denote the finite inte- grals resulting from this integration, in order to distinguish them from the preceding finite integrals, which refer to the aggregate of all the bodies of the system. The integral of mP/dx + A.dx) is evidently equal to JinP.dx f therefore we shall have const.— dx*+dy' + dz» ^^ [ ( ^ dx* ^, / dy* \ , dz\-i* ^■''- IP +^'-'"r^-5F)+^^-ir) + (--^)( —21.fm.(P'dx + Q.dp + R.dz) ; dx * In this equation, though the value of A. — — may be finite, still dx-\-i^.(Ix, and the variation of the time may be indefinitely small, and V integrating with respect to this C dx dx "J dx^ quantity, 2,2.»n. I ——.A.— V = 2.>n. —j-j-, or it may be otherwise expressed thus, A.(x') :;^(see Lacroix No. 344) 2xh-{-h'^, and if /« be made equal to Ax, it becomes 2:r.Ax-|-(A.x)*, •.• 2. 2.Xj;A..r+(A.x)*) = 2,.(2a:.A.l-l-(Aj;)= )-(- 2,.(Ax)« — x^ + 2,.(Ax4), consequently, if we multiply the preceding equation by two, and substitute dx in place of r, and then integrate, we obtain the expression which has been given in the text. *ia CELESTIAL MECHANICS, therefore v, v, v" denoting the velocities of m, m', in!', &c. we shall have s.^.'=const.-E,2.«.UA.^r+ { A.^r4- u.^xx ' (^ dt f \ dt S '- dt S i + 2I.j:m.{P.dx+Q.dj/ +R.dz'). The quantity contained under the sign Z^, being necessarily positive, we may perceive that tlie living force of the system is diminished by the mutual action of the bodies, as often as during the motion, any of the variations A.— — ,A.— ^, &c. are finite. Moreover, the preceding Civ at equation affords a simple means of determining the quantity of this diminution. At each abrupt variation of the motion of the system,* the velocity * At every abrupt change in the motion of the system, the velocity is not always di- minished for every body, but the expression which is here given may be considered as general, by supposing that when the velocity is increased, a negative portion of it has beeti destroyed, and the square of the velocity after the shock is equal to ^■'"' dt'' and as ^ 2.dxi\.dx + '2{A.dxY + 2.dy^.dy + 2{A.dyy +2dz:^.dz+2{A.dz)*, =0, by subtracting this equation from the preceding, we obtain the square of the velocity after the shock, equal to .„ (dx'+dy'+dz-) ^___ (S.dx)^MM>/y-^(^-^=:)' dt'' — "• • dt' and as the square of the velocity before the shock is equal to l.mv'' = 2.W. — ^ — ■^ ^, the square of the velocity lost by the shock =2.ot. F* _v« Jj^±y±{±_clyy+(A.dz)_' . dt' ' consequently the loss which the living forces experierjce, is equal to the sum of tlie living forces, which would belong to the system, if each body was actuated by that velocity which it loses by the shock. PART I.— BOOK I. 119 of m, may be conceived to be resolved into two others, of which one v subsists in the following instant, the other V being destroyed by the action of the other bodies, but the velocity of vi before the decomposi- . , . \/dx* + dy*-{-dz*, ■, , . n tion being -^ , and changing afterwards into dt it is easy to perceive that ( dt S t dt S c dt S ' consequently the preceding equation may be made to assume the fol- lowing form, 2.m'« = const.— 2^.2.7??. V'—2.1.fm.{P.dx + Q.di/ + .dz),* * The variation of the vis viva of the system, is equal to 22m,{P.dx-{-Q,.di/+ R.dz) consequently when this expression vanishes, i. e. when f/.2.(mt'*) vanishes, the vis viva of the system, equal to 2.()??u=), is a maximum, or a minimum; but it appears from the principle of virtual velocities, that 2m.(P.Sx-[- Q.?!/-\-R.h) is equal to nothing, when the forces P, Q, R, P", constitute an equilibrium ; and since the differentials dx, dy, dz, may be substituted for the variations 'hx, tij, S^, when they are subjected to satisfy the condi- tions of the connection of the parts of the system, l.m.[P.dx-\- Q,.dy-\-R.dz) is equal to nothing, in the same circumstances ; •.• when the forces P, Q, R, Pi, constitute an equi- librium, the vis viva of the system is a maximum or a minimum. And as it appears from note to page 96, that the positions of equiUbrium of a system of heavj' bodies, correspond to the instants, when the centre of gravity is the highest or lowest possible, the sum of the living forces is always a maximum or a minimum when the centre ceases to ascend, antl commences to descend, and when it ceases to descend and commences to ascend. The value of the vis viva is a minimum in the first case, and a maximum in the second, for 'Zm.{P.dx-\-Q.dy-{-R.dz) corresponds to the expression S.Ji-f- S'.Ss'-f S'''3i''''-|-&c. in page 96, and •.• by substitution we have Imv'^ = c-\-s,.S'^m. con- sequently 5.?)!ii» is a maximum or minimum, when s, is a maximum or minimum. When S.mii* . is a maximum, the equilibrium is stable ; when a minimum, the equilibrium is in- stable. For from the definition of stability, (see No. 28) it appears that if the system is only agitated Sy one sole species of simple oscillation, the bodies composing it will perpe- 120 CELESTIAL MECAHNICS, 20. If in the equation (P) of No. 18, we suppose, Sxf =z Sx-\-ix/ ; Sy' = Sy-\-Sy; ; Sz' zz Sz-\-Sz; ; tually tend to revert to the position of equilibrium, consequently their ve'ocities will di- minish according as their distance from the position of equilibrium is increased, and •.- tlie sign of the second differential of ^ will be negative, consequently 2 wd^. will be a maximum in this case ; and it may be shewn by a like process of reasoning, that the vis viva of the system is a minimum, when the equilibrium is instable. From a comparison of this observation with the note to page 96, it appears that in a system of heavy bodies, when the vis viva is a maximum, the centre of gravity is the lowest possible, and highest when the vis viva is a minunum. This may be more strictly demonstrated thus : if the system be disturbed by an indefinitely small quantity from the position of equilibrium, by substituting for P, Q,R, P', &c. their values in terms of the coordinates, and then expanding the resulting expressioninto a series ascend- ing according to the variations of these coordinates, the first term of the series will be the value of tp, when the system is in equUibrio ; and since it is given, it may be made to coalesce with the constant quantity c, which was introduced by the integration ; the second term va- nishes by the conditions of the problem ; and when 2.»nv" . is a maximum, the theory of max- ima and minima shews that the third term of the expansion may be made to assume the form of a sum of squares, affected with a negative sign, see Locrobc, No.lSl; the number of terms in this sum, being equal to the number of variations, or independant variables ; the terms whose squares we have assumed, ai'e linear functions of the variations of the coordinates, and vanish at the same time with them ; they are therefore greater than the sum of all the remaining terms of the expansion. The constant quantity being equal to the sum of c, and of the value of 'S.mv'^. when the forces P,\Q, R, P', &c. constitute an equilibrium, it is necessarily positive, and may be rendered as small as small as we please, by dimi- nishing the velocities ; but it is always greater than the greatest of the quantities whose squares have been substituted in place of the variations of the coordinates ; for if it were less, this quantity being negative, would exceed the constant quantity, and therefore render the value of S.mi)'. negative, consequently these squares, and the variations of the coordi- nates, of which they are linear functions, must always remain very small, v the system will always oscillate about the position of equilibrium, and this equilibrium will be stable. But in the case of a minimum it is not requisite that the variations should be always constrained to be very small, in order to satisfy the equation of living forces when <p is a minimum ; this, indeed, does not prove that there is no limit then to these variations which is necessary, in order that the equilibrium may be instable ; in order to shew this we should substitute for these variations, their values in functions [of the time, and then shew from the form of these functions, that they increase indefinitely with the time, however small the primi- PART L— BOOK I. 121 ix" = Jx + Sx',' ; Sy"=Sy+hj," j^s''^ Sz + Sz;' •* &c. by substituting these variations, in the expressions of the variations Sf, Sf', Sf, &c. of the mutual distances of the bodies composing the system, the vahies of which have been given in No. 15; we shall find that the variations Sx, Sy, Sz, will disappear from those expressions. If the system be free, that is, if it have none of its parts connected with foreign bodies, the conditions relative to the mutual connection of the bodies, will only depend on their mutual distances, and therefore the variations S^, Sy, Sz, will be independent of these conditions ; conse- quently when we substitute in place of Sx', Sy', Sz', Sx,", &c. their pre- ceding values in the equation (P), we should put the coefficients of the R live velocities may be. For a complete solution of the problem of the small oscillations of a system, the reader is referred to the Mechanique Analytique of Lagrange, 5th and 6th section, seconde partie, where the important problem of coexisting oscillations is discussed in all its generality, and all difficulties are cleared up ; see also Notes to No. 23 and SO, of this book. * It is ahvays possible to make these substitutions, for it in fact comes to transferring tlie origin of the coordinates to a point of which the coordinates are equal to x, y, z, res- pectively ; as the expression for y, _ (j;'—x).(g.r'—?x}+0/— »)■(?/— ?;y)+(z'—z).' iz'—2z) ~ f equal by substituting for x', y' , r', Ix' , h/, h', their values, f ^ f consequently as 3x, Sy, Js, disappear from the expressions of the variations \f, If', and as when the s3stem is at liberty, the conditions relating to the mutual connexion of its parts, depend only on their distance from each other, the variations Jx, Sy, Sz, will be inde- pendent of these conditions, ■.• substituting for Sx', 5y', ^z' in the equation (PJ, the values which have been just given for them, the coefficients Ix, ly, ^z, must be put equal to nothing. 12-2 CELESTIAL MECHANICS, variations Sx, Sy, Sz, separately equal to nothing ; which gives the three following equations : Let us suppose that A', Y, Z, are the three coordinates of the centre of gravity of the system ; by No. 15 we shall have, ■y_s.nix _ -^ _ T.V11/ s..mz -A j X — ; ii = . ; •s-.m s.m s.m consequently 0 = ^_^f!£ . o = ^'^_^-^Q . o = ^ s.?«-R ^ (If S.??2 ' </f S.TO ' ~ df '~'s..m ' therefore the motion of the centre of gravity of the system is the same • By actually substituting for Sx', Sy, "^J, Iz", &c. in the equation (P) we obtain 0=. +•■'■>'■ {■'^-^\^--'':-{'-^- p} the terms in this expression which are multiplied by Sx, Sy, Sz, respectively, are by adding them together and being independent of the conditions of the connection of the system, they must be put ieverally equal to nothing. t SuiceX= , Y= —, &c. -r5- = 2.7W.-— = , 2.WJ 2.m di' dt- 2.?k 2to because ^.m.-—^ — 2m.P=0. PART I.— BOOK I. 123 as if all the bodies m, m', &c. were concentrated in this point, the forces which solicit the system being applied to it. If the system is only subjected to the mutual action of the bodies which compose it, and to their reciprocal attractions, we shall have 0 = -z.mP ; 0 = -z.mQ ; O = -z.mR j for p designating the reciprocal action of m on m', whatever its nature may be, and y' denoting the mutual distance of these two bodies ; we shall have, in consequence of this sole action, „ (x — x) „ (y — y') p [z — z') mP=.p.^^—^ — -; mQ=p. ^- . '•,mR=p. - — ■, — - ; mF = p. ~ . ; mQ=p. '--^ •^- ; mR=p. ^ ; from which we collect Oz=mP+m'P' ; 0=mQ + m'Q ; 0-mR + m'R'; * and it is evident that these equations obtain, even in the case in It 2 V • / — X, y — ■)/, z' — z, being the coordinates of nt relative to the new origin of the forces, and the action of p being directed along the line the part of mP, which corresponds to the force p resolved parallel to the axis of x=p — , the analogous parts of otQ, and mR, axe p. \. >p- respectively, in like manner the forces soliciting m' parallel to the coordinates, arising from the action of p, -p. J ,P- J- 'P- f > .-. when the sole force soliciting tn and m' arises from p, which expresses the reciprocal action •f m on m', we have mP-^-m'P',—p\JI' ' — =0. Action being equal to reaction, and its direction being contrary thereto, when two bor 124 CELESTIAL MECHANICS, which the bodies exercise on each other, a finite action in an instant. Their reciprocal action disappears from the integrals ^.mP, ^.mQ, 'Z.mR, and consequently, these expressions vanish, when the system is not so- licited by any extraneous forces. In this case we have and by integrating X^a + bt: r~a'-{-b't; Z=a"+b"t;* a, b, a', b', a", b", being constant arbitrary quantities. By eliminating the time t, we shall have an equation of the first order, between either X and Y, or X and Z ; consequently the motion of the centre of gravity is rectilinear. Moreover, its velocity being equal to v/{?F^ If F^ {f }■ or to v' b b'--^b'\ it is constant, and the motion is uniform. It is manifest, from the preceding analysis, that this invariability of the motion of the centre of gravity of a system of bodies, whatever their mutual action may bc,t subsists even in the case in which any one dies concurring, exercise on each other a finite action in au instant, their reciprocal action will disappear in the expressions S.jwP, 5.mQ, &c. in fact, as we can always suppose the action of the bodies to be effected by means of a spring, interposed between them, which endtavours to restore itself after the shock, the effect of tlie shoclc will be produced by force* of tlie same nature with ;;, which, as we have seen, disappear in the expressions 'S.mP, S.otQ, S.mii. ♦ By integrating once we get — = b, .: dX:z bdt, and X~ ht+a; the constant quantities Clt a, a', a", are equal to the coordinates of the centre of gravity when / ■=. 0, and b, V , b", are equal to the velocity of the centre of gravity resolved parallel to the coordinates. See notes to pageSl. ■\ In fact, from what has been observed, in the note to page 116, it is evident that the principle of D'Alembert is true, whether the velocities acquired by the bodies be finite, after a given time, or indefinitely small, or whether the velocities be partly finite, and partly infinitely small, such as arise from the action of accelerating forces, and both PART I.— BOOK I. 125 of the bodies loses in an instant, by this action, a finite quantity of motion.* 21. If we make Sjif-=z ^ + Sx ; Sx =^ h^< ; &c. y y Sy= + Sy/,Sij'= + Sy ; Sy'= + Sy;' ; &c. t y y y , tiie variation J'.r will again disappear from the expressions iJ/, ^', ^f", &c. ; therefore, by supposing the system free, the conditions relative before and afler the impact, we have 0= Trj-> 0= -rr^t *c. and also — . S.m = (/<» ' df^ ' ' dt dx . . . ■ 2.m. — , &c, = the quantity of motion, and sinc^' ' by hypothesis the quantity of motion dx lost, equal to the difference between Sm.— before and after impact, should be = to nothing, such as would cause an equilibrium in the system, it follows that -t^.S.jb. before and after impact must be the same, but 2.?n being given, — equal to the velocity of tlie centre of gravity, will be the same before and after impact. * As the centre of gravity of a system, moves in the same manner as a body equal to the sum of the bodies would move, if placed in the centre of gravity, provided that the same momenta were communicated to it, which are impressed on the respective bodies of the system, the motion and direction cf the centre of gravity, may be always determined by tlic law of composition of forces. If the several bodies of a system were only subjected to their mutual action, then they would meet in the centre of gravity, for the bodies must meet, and the centre of gravity remains at rest. t The fractional part of these expressions for Sy, Sx'/, Sy, ?/, Sj/'^, &c. arises from the ro- tatory motion of the system about an axis parallel to z, for it appears from Nos. 22 and 25, that when the direction of the impulse does not pass througli tiie centre of gravity, the body acquires both a rotatory and rectilinear motion, now if the only motion impressed on the ' system was that of rotation, then the element of the angle described by the body m, is equal to the variation of the sine divided by the cosine =- — ^. Sj, the elementary angle de scribed by 126 CELESTIAL MECHANICS, to tlie connection of the parts of the system will only influence the va- riations Sf, Sf" &c. ; the variation Sx is independent of them, and entirely arbitrary ; thus by substituting in the equation (P) of No. 1 8, in the place of Sx, $x", Sx"\ &c. Sy, Sy", Sf, &c. their preceding values. = \/£2+^ ^'^J''+^/' . _"^^-'^"'+.y'* . ix- .'. the variation of j^ will be equal to t:.^.. v^'g'^-)-y'~ v' V ^ ' — , Sj,=- . dx the same may be proved of the other variations ix', 3x'' y ^/x'^^-y^ y ^ ^ ' ^^ ' +y' = the distance of m from the axis of r, .*. , ,- is equal to the sine of tlie angle which Vx^+i/' makes with y. If the expression — i— — -^ be consi- y a: dered with respect to the cosine^, the variation o_y = — ?x. ~ "^-^ . y Vx^'+^: ix.X „ , . ) tor the variation of the cosme is equal to the variation of the arc affected with a negative sign, and divided by the sine, and as the variation of the angle described hr , \/x--\-y"' ... . . Vx^^'+Z'^ "' -^ . ox, this expression being referred to the cosine is equal to -^ — .• . 3x.= . 3x. If in the expression we substitute for ^x', ^.i", Sj/, Sy', Sy , &-c. their values, it becomes f 3(i/''ix xi/.ox , » . » , » , » , y'x.'.dx y.x''^x . y'xHx y y y y y ~-—+yhl—y-h!~y h.^y^yrrf^ j^ therefore the variation Jx disappears from the expressions "if, "if, &'C. Making the same substitutions in the equation (P) it becomes PART I.—BOOK I. 127 we should put the coefficient of Sx separately equal to nothing, which gives 0='SM. ^^ ^~~^—^-+-z..m. {Pt/ — Qx) ; from which we deduce by integrating with respect to the time t, c=z.m. C-^^i^— -V^-^) -|-s>?.(Py— Qx). dt; c being a constant arbitrary quantity. In this integral, we may change the coordinates y, i/, &c. into z, s', provided that we substitute in place of the forces Q, Q, &c. parallel to the axis of ^, the forces B, R', 2)arallel to the axis of z, which gives, d = z.m. (^dz-zclj/) ^ s/.m.(P3— i?^). di ; <f being a new arbitrary quantity. In like manner we shall have c"=x.m. C3/^~— 3^^V) +s,f,m.(Qz—Ry). dt; 1 c" being a third arbitrary quantity. o-,4.,.g-P^-i.4S-«}+/..{--p.| I "*■ I d- — !'+'"• \ J \—mPy\-mQ,x—m'P'i/^m'Q^3/,&c. therefore if tliis expression is extended to all the coordinates, it will become 128 CELESTIAL MECHANICS, Let us suppose, that the bodies of the system are only subjected to their mutual action, and to a force directed towards the origin of the coordinates. Let ^; denote, as before, the reciprocal action of m on m', wc shall have in consequence of this sole action, 0-m.(Pij—Qx) + 7n'.{Fy'—Qx') ; thus the mutual action of the bodies disappears from the finite integral ^.nu(^P7/ — Q.r). Let 5' be the force which solicits m towards the origin of tlie coordinates ; in consequence of this sole force, we shall have P- . -^'^ -: Q= -'^^^ consequently the force S disappears from the expression Pi/ — Qx, thus, in the case in which the different bodies composing the system are only solicited by their action and mutual attraction, and by forces directed towards the origin of the coordinates, we have c = ^.m.— — - — - — ~ ; c —s.m.-^ ; -; c =l.m. -^ ^^^ dt . dt dt If we project the body m, on the plane of :r and of 3/, the differential — "^^ — , will represent the area which the radius vector, drawn from the origin of the coordinates to the projection of m, describes in the time dt ; consequently the sum of the areas, multiplied respectively by the masses of the bodies, is proportional to the element of the time, fron.i which it follows, that in a finite time, it is proportional to the time. I( is this which constitutes the principle of the conservation of areas.* ♦ When the bodies are only subjected to their reciprocal action, 2.OT.(Py— Qr)=w.(Py— Qx)+m'. (F/— QV)+ &c. — by substituting for m P, m Q, their values, given in page 122, ^{il/—x't/—)/l-\-xi/) 4. .ry— r;/— ya'-f-j'y) 7 _^ PART I.— BOOK I. 129 The fixed plane of x and of y being arbitrary, this principle obtains for any plane whatever, and if the force S vanishes, i. e. if the bodies are only subjected to their reciprocal action and mutual attraction, the origin of the coordinates is arbitrary, and may be in any point whatever. Finally, it is evident from what precedes, that this principle subsists, even when by the mutual action of the bodies composing the system, they undergo sudden changes in their motions. There exists a plane, with respect to which c and c" vanish, and which, for this reason, it is interesting to know, for it is manifest that see preceding number. If the bodies are solicited by forces directed towards a fixed point, then making this point the origin of tlie coordinates, consequently this force will also disappear from the expression Py — Qx, .•. in these two , xdii — udx xdii — iidx , . , , . „ cases we nave c = 2 m. — , ac ; — — - iz the area wluch the projection of the radius vector on the plane ■ of x, y, describes in the time dt, see notes to No. 6, page 27. Z.m.{Py — Qx) ~ 0, also when F and Q, &c. vanish, i. e. when the system is not actuated by any accelerating force, but only moved by an initial impulse; .•. the principle of the conservation of the areas obtains in these three cases; 1st. when the forces are only the result of the mutual action of the bodies composing the system; 2ndly, when the forces pass through the origin of the coordinates ; and 3dly, when the system is moved by a primitive impulse. In the first and last case, the origin of the coordinates may be any point whatever. If there is ajixed point in the system, the equations {Z) are • only true when this point is made the origin of the coordinates, any other point being made the origin, the moment Py — Qx will not disappear, see notes to No. 3, page 12 ; if •.■ in these circumstances the bodies are solicited by forces directed towards a given centre, this centre coincides with the fixed point of the system, when the equations (Z) obtain ; if there are two fixed points in the system, only one of the equations (Z) will sub- sist, to wit, that which contains those coordinates, the plane of which is perpendicular to line joining the given points, the origin of the coordinates may be any point whatever in tliis line, see notes to No. 15, page 88. The constant quantities c, d, c", may be determined at any instant, when the velocities and the coordinates of the bodies of the system, are given at that instant. * 130 CELESTIAL MECHANICS, the equality of c' and c" to nothing, ought to simpUfy considerably the investigation of the motion of a system of bodies. In order to de- termine this plane, we must refer the coordinates .v,?/,^, to three other axes having the same origin as the preceding. Let there- fore 9 represent the inclination of the required plane, formed by two of the new axes, with the plane of a:' and oiy, and ^ the angle which the axis of x constitutes with the intersection of these two planes, so that S may be the inclination of the third new axis with the plane of x and of y, and ^ may represent the angle which its projection on the same plane, makes with the axis of x, ir being the semi periphery. In order to assist the imagination, let us suppose the origin of the coordinates to be at the centre of the earth ; and that the plane of x and of y coincides with the plane of the ecliptic, and that the axis of z is the line drawn from the centre of the earth to the north pole of the ecliptic : moreover, let us suppose that the required plane is that of the equator, and that the third new axis, is the axis of rotation of the earth, directed towards the north pole ; 0 will represent the obliquity of the ecliptic, and 4 will be the longitude of the fixed axis of x, relative to the moveable equinox of spring. The two first new axes will be in the plane of the equator, and by calling (p, the angular distance of the first of those axes from this equinox, <p will represent the rotation of the earth rec- koned from the same equinox, and —-{-?> will be the angular distance of the second of these axes from the same equinox. We will name these three new axes, principal axes. Let.r,,_y^, 2, represent the coordinates of m referred, first to the line drawn from the origin of the coordinates, to the- equinox of spring ; x being reckoned positive on this side of the equinox ; 2dly, to the pro- jection of the third principal axis on the plane of x and of y ; Sdly to the axis of z, we shall have PART I.— BOOK I. 131 xz=.xi. COS. ^-\-y,. sin. 4* » y=j/,. COS. v|/— T^ sin. 4/ ;• » ^ */• Let j,^ y,t, Zii, be the coordinates referred, 1st to the line of the equi- nox of spring ; 2dly, to the perpendicular to this line in the plane of the equator ; Sdly, to the third principal axis ; we shall have ^1 = x„ \ y, = y,!. cos. %-\-Z/,. sin. 6 ; z^ = Zi,. cos. 6—3/,,. sin. 6. Finally, let x,„, y ^^„ ^„,, be the cooordinates of m, referred to the first, • s 2 • As the axes of the coordinates .r,, y., exist in the plane of x, y, and as the angle wliich the axis of x makes with the axis of x_, is equal to -i^, we have by the knowii formula; for the transformation of one system of rectangular coordinates, into another system existing in the same plane, x=x,. cos. ■\'-\-yi- sin. •4' ;yr:_y,. cos. -i^ — x. sin. ij/ ; and because the axis of 2 coincides with the axis of s , we have z=z. Comparing the coordinates, * ,^,,z,, with the coordinates x^,y^^,z^, it appears that the axis of .Ty coincides with the axis of x^,, and consequently x,=t// ; and as the axis of »/, is in the plane of the ecliptic, perpendicular to the line of equinox of spring, and aa the axis of ^,, exists in the plane of the equator perpendicular to the same line, it is manifest that the angle formed by these axes is equal to the angle i, the inclination of the two planes, and that these two lines and the axes of z^ and z^^, which are respectively perpendicular to those planes, exist in the same plane, consequently we have, as before, yr^Vii' '^o*' ^-\-'^ir s'"- *> -/=-//• cos. i. — _!/„ sin. i. Lastly, it appears that the axis of z,i coincides with the axis of s,,^, and consequently that z,,'=z,i,, ; and as the axis of x„ and 2/,„ and of x,,^ and^„, are in the plane of equator; and as by hypothesis, ij) is equal to the angle which the axis of*,,, makes with the line of equinox of spring, which line is supposed to coincide with the axis of x^,, we have x^,=x^„. cos. <p — ;y„^. sin.?; yi,=y,„. COS. ip4-^,„- *'"■ ?• % substituting for x^ y^, x,^ y,„ their values, we obtain x=x,. cos. ■>H".y/- sin- ^ = (-r//- cos. ■^■^y,,- cos. 6. sin. ilz+z,,. sin. «. sin. 4-) = (r„,. cos. *. COS. •vj/ — ly,^^. cos. -vj/. sin. <p-\-y„,- cos. i. sin. ^. cos. (p. 4"^,„> cos, t. sin.^/. sin. (f -[-z,,,. sin 6. sin. 4"), •." by concinnating we obtain *=x„^(cos. t. sui. 4'. sin. ip+cos. (p. cos. ■^) -\-y„, (cos. i. sin. 4'. cos. ip. — cos. 4'- sin. ?)-J-z^,^. sin. i. sin. i^, which is the expression given in the text ; by a similar process we could derive values for y and z. 132 CELESTIAL MECHANICS, second, and third principal axes ; we shall have x^ = x^^. COS. (p — 1/ , . sin. ip ; From which it is easy to deduce X = «'/^/.(cos. 9. sin. vf/. sin. ip + cos. ^. cos. (p) + y^^,.{cos. 6. sin. ■^. cos. ip — cos. vj/. sin. ip) 5;^,/. (sin. 0. sin. 4^) ; «/ =07/^^. (cos. 6. cos. ■]>. sin. (p — sin. \|/. cos. (p) + ^//^.(cos. G. cos. 4/. cos. 9 + sin. i}/. sin. (f) +z^^^ (sin. 6. cos. ^'); z = z,^. COS. 0 — j/j,^. sin. 6. cos. <p — o:*^^^ sin, 9. sin. (p.* If we multiply these values o{ :r, y, z, by the respective coefficients of ♦ If any line x is drawn from the origin of the coordinates x,,,, y^^, z,,^, and if A, B, C, represent the cosines of the angles which a- makes with z_,,, ^/,^,, 2,,,, respectively, then X = Axiii-\-Byiii + C'2„^, for if a perpendicular erected from the extremity of x meets a X line r, whose coordinates are x,^,, y,,,, s,,,, then — is equal to the cosine of the angle which X makes with r, and — ^, ^^, ^^, are equal to the cosines of the angles which r r r r makes with x„,, ^,,„ z„„ •.• we have by note to page 7, — = ^. — ' + B. — r r r z + C.-^, :• x=zAx^i^-\-By^^^+Cz^^^. Consequently we infer that the coefficients of x^^^, Villi ''■IIP '** ^^ expression given in the text for x, y, z, are equal to the cosines of the angles which the axis of x, y, z, make with the principal axes respectively ; therefore sin. t. MD. >}', is equal to the cosine of the angle which the axis of 2„,, makes with the axis of PART L—BOOK I. 133 ■Xw, in the preceding expressions ; we shall have, by adding them toge- ther, x^^^ = a:. (COS. 9. sin. 4/. sin. p+cos. v|/. cos. <p) + 3/.(cos. 9. COS. vj/. sin. <p — sin. v}/. cos. <p) — z, sin. 9. sin. <p. By multiplying in like manner the values of x, y, z, by the respec- tive coefficients of i/,„ in the same expression, and afterwards by the co- efficients of 5,„, we shall have z/,,y = a;.(cos. 9. sin. v}/. cos. ^— cos. vf/. sin. ip.)4- X, :• equal to the cosine of the angle which the plane of ?/,,,, j ^ , makes with the plane y, z; in like manner sin. S. cos. ip, is equal to the cosine of the angle contained between the axis of z^^^ and of^,=the cosine of the inclination of the plane a;,,^, y,^^ to the plane s, z, also sin. i. sin. ip, sin. 6, cos. (p, cos. i. are equal to the cosines of the angles which the axis of z makes with the axes of x,„, i/,„, and z,„, respectively, see No. 27. We may observe that in the general expressions for the transformations of one system to another of rectangular coordinates, which are of the following form : ^ = ^x,„-\-Bt/,„+Cz„„ z= A„.x,i,-\-B/j.y,„-\-C„.z,,„ there are six equations of condition, i. e. A* + A,^ + A,/,=l AB+A,B,+A,yB// =0, B^ +B,' + B,/, =1 A C+Af,+A,,C„ =0, C^ + C; + C,,/,=l BC+B,C,-\-B,,C„=:0, ■which are derived from the identity between the expressions a;* +y^ +z», and x,,,'' H-i/,„--|- z,,,*, for they are respectively equal to the square of the distance of the same point, from the common origin of the coordinates, •.• three of the nine coefficients which are intro duced by the transformation, may be regarded as undetermined; these three undeter- mined quantities are, in fact, the angles 6, ij/, and (p ; for, by substituting in the six preceding equations of condition for A, B, C, A,, &c. their values in functions of the angles 6, 4^, and (p, the resulting equations will become identical, and there arises no relation between f, 4^, and p. 1S4 CELESTIAL MECHANICS, ^.(cos. 0. COS. \J/, COS. ^ + sin. \|/. sin. (p)^z. sin. 0. cos. ^ ; s «„, =*. sin. 0. sin. ^^+2/- sin. 0. cos. tj/. + 2. cos. 6. These different transformations will be very useful hereafter, we will obtain the coordinates corresponding to the bodies m, m", &c. : by placing one, two, &c. marks above the coordinates x^, y^, z,^^, y ^^^ z,,.* • If we actually perform this operation we shall obtain r.(cos. «. sin. 4-. sin. (p + cos. 4- cos. ?i)=x,^,.(cos. -6. sin. s^-. sin. 2^+cos. ^■^, cos. V + 2 cos. d. sin. T^. COS. ^. sin. ip. cos. (p.) ■\-t/f,f.(coa. 'e. sin. -•vj'. sin.ip.cos. ip+cos.«.sin. 4-. cos 4. cos.*^ — cos. ^.sin.i^. cos. 4'. sin. V — cos. '■4- s'"^- 'P- cos. Ip.) -f-z,,,.(siii. 6. cos. (I. sin. '-4/. sin. (p + sin. 6. sin. ■<}/. cos. •vj'. cos. <p) ; ^.(cos. 6. cos. 4. sin. ? — sin. if. cos. *.) ;ex^„.(cos, <. COS. '4/. sin. *<fi+sin. ^^.cos. "(p — 2(cos. 6. sin. 4/. cos. ■4'- sin. ?. cos. $) +y„,.(cos. *6, cos. 'i^. sin, ^. cos. ip+cos. «. sin. -vf- cos. ■vj'. sin. •<? — cos. i, sin, ■vj/. cos, -i-. cos. "<p — sin. ^■^. sin. ^. cos. <p.) -j-s„^.(sin. 6. cos. «. cos. 'i^- sin. $> — sin. 6. sin. ■i|'. cos. 4-. cos. ^) — r. sin. I. sin. <p= — 2,,^. sin, d. cos. d. sin. ip+y„,, sin. *1 sin. <p. cos. ^+x,,,. sin. s^. s;n. '^ ; adding these three equations together, and making the terms which are at the right hand side to coalesce, we shall get the coefficients of J',,,= to cos. '<». sin. ":p+cos '?.+ sin. '*. sin. 'ip, (=sin. ^ip — sin. ^6. sin. ^^ + cos. '(f+sin, ^6. sin. *V) =1> t'le coefficients of y^,, will be equal to cos. ^6. sin. (p. cos. $r-sin. ip. cos. ^-j-sin. ^$. sin. ij. cos. ip:=0, in like manner the coefficient of z^^ =sin. 6. cos. 6. sin. (p — sin. 6. cos. d. sin. tp^O ; the terms at the other side are those which have been given in the text. In like manner to obtain the value of y^,i, a corresponding multiplication gives x.(cos. (. sin. ■^. cos. ^ — cos. i^. sin. f) = x^,^.(cos.*^. sin. *4/,sin. ^.cos. ?i-j-cos. *. sin. 4/. cos. 4. cos. 'f — cos. 6. sin, 4. cos. -v^. sin. *f— cos, 'if- sin. (p. COS. (p) •]-yf,J[cos, "e. sin. ^T^. COS. '(p-J-cos. ^•4, sin. -?i— 2. cos. 6. sin. 4. cos. ^. sin. f. cos. ^\ + z,„(sin. ). cos. e. sin. ^il'- cos. <p — sin 6. sin. ■4'. cos. •4-. sin. (?) PART I.— BOOK I. Its From what precedes, it is easy to conclude, by substituting c, C, c", in place of dt dt dt * t/.(cos. e. COS. ip- COS. <p-|-sin. ^|'. sin. ip) = :c,„(. COS. H, COS. ^^z. sin. ^. cos. <p — cos. 6. sin. ■v^. cos. %//. cos. ^^ -j-cos. 6, sin. 4'' cos. i^. sin.-^ — sin. ^. sin. <p. cos. <p) +y„,(cos. ^. cos. 'if', cos. 'ip+sin. ^. sin. 'ip + 2. cos. 9. sin. (p. cos. ^. sin. 4". cos. ■vf-) -f-«,„' sin. *. cos. i. cos. -i^. cos. if+sin. 6. sin.i/^. cos. i|/. sin. f.) — z. sin. S. cos. ip=r — 1,„. sin. i. cos. S. cos. ip+y/,/- sin. ^i. cos. "ip+x,„. sin. ^6. sin. ^. cos. ?, -aJding those quantities together, and concinnating as before, we obtain j.(cos. i. sin. ■4'- cos. (p — cos. ■4'. sin. ^)4-^-(cos. 6, cos. tp. cos. p + sin. •\^. sin. 9) — z. sin. ^. cos. ip— «,„.(cos. ^t. sin. Ip. cos: ip — sin. (p. cos. <l>-{-sin. ^6. sin. (p. cos. 9)=0, +j/„^(cos. 2^. cos. *?i+sin. ^^-f sin. ^i. cos. *(p)=r^„,, + r„,. (sin. 9. cos. *. cos. <p — sin. «. cos. i- cos. ip)=.0. Fw the value of a,,^, by performing similar operations, we obtain x. sin. i. sin. 4"= c,„.(sin. 6. COS. 4. sin. ^. sin. (p-{-gin. 0. sin. ij/. cos. 4'* cos- f ) +y//,.(sin. t. COS. <. sin. ^^J'* cos. $ — sin. e. sin. ■^. cos. i^. sin. ip) +£^„.(sin. ^*. sin. ^i/'' y. sin. #. cos. \f'=: «„,.(sin. i. cos. *. cos. ^. sin. ip — sin. 0. sin. i^. cos. i^- cos. ip) +^///'(sin. «. COS. (. cos, 24,. cos. 9 + 8in. «. sin. 4. cos. ij/. sin. ^)-f.a,_^(8in. '*. cos. '-vf-.) «. cos. 6=. — X,,,. sin. e. cos. ^. sin. <p — ^,,,.(sin. 6. cos. ^. cos. ?+«;,,• (cos. *»), <,*. adding tlie corresponding quantities together, we obtain 136 CELESTIAL MECHANICS, that zm.'" •^^"'~~^^"' •'""'■ =c. cos. 6 — c'. sin. «. cos ij^ + c". sin. t. sin 4' ; dt X .dz . — z .dx sm. -^^^^ — '" , "^ — ^ =c. sm. fi. COS. (? * dt ■ X. (sin. i. sin. ■4')+^' sin. i. cos. 4" +-• cos. «= *,,^.(sin. i. COS. C. sin. ?) — sin. ^. cos. i. sin.if).)=0, H-y,„.(sin. ^. COS. i. COS. ip — sin. C. cos. i, cos. ?>)=0, +3„,.(sin. s^.+cos. -«)=3„^. * When we substitute for the expression Xndym — yuM^u,' the respective values of jr„,, 'i^iii'y,,!' ^Viii' "^ functions of *, dx,y, dy, and of the angles 6,-^, and (p, it is not ne- cessary to take into account any expression, in which the variable part is the product ot a coordinate into its own differential, because this expression occurs again, affected witl% a sign, the opposite to that, with which it was affected before. By performing the pre- scribed multiplication of the value of a:,,, into the value of c/y,,, of y,,^ into dx„, we obtain x„_-dy ii,-=xdy.[cos. -6. sin. -ij/. cos »|/. sin. <p. cos. $-}-cos. 6. sin. "■•p- sin. '<? "h cos. i. cos.'i^. COS. ^^+sin. i|/. cos. 4'. sin. (p. cos. (f), -J-y.(/x.(cos. "6, sin. i|^. cos. ■vf-. sin. ip. cos. <p — cos. 6. cos. ^-vj/. sin. '^ — cos. ). sin. ^i)/- cos. ^?i-|-sin. i|/. cos. ■<i^. sin. ip. cos. ^), — z.c?x.(sin. 6. COS. ^. sin. ij/. sin. ?>. cos. (p — sin. 6. cos. ^f/. sin. "ip), — 3.rfy.(sin. ^. COS. 6. COS. ij'. sin. ifi. cos. ?> + sin. «. sin. ■4'. sin. '<p), — j.t/z.(sin. 6. COS. «. sin.-|. sin. (f>. cos. ?)-j-sin. 6. cos. •J', cos. -ip), — y.dz.{siu. 6. cos. S. cos. ■^. sin. ^. cos. (p — sin. 6. sin. t^. cos. "if), y^^,.rfr,„.rr«.rf_y.(cos. 2«. sin. il/- cos. il/. sin. (p. cos. ip — cos. 6. sin. s-J.. cos. *ip — cos, t. cos. yvf^. sin. -.p+sin. i^, cos. i^. sin. (p. cos. ip), -}-y.rfx.(cos, 2«. sin. ij/, cos. -J/, sin. (p. cos. <p-f-cos. 6. cos. '4- cos. -ip -J- cos. i. sin. 24. sin. 2(p-|-sin. i^. cos. if'- sin. (p. cos. ?), — ^z.</j;.(sin. S. cos. 6. sin, if- sin. (J. cos. (p-fsin. 6. cos. 4''C0S. ^^), — z.dy.(sm. 6. COS. tf. cos. 4. sin. ip- cos. 9 — sin. «. sin. 4. cos. ^^), — x.rfr.(siii. 6. COS. «. sin. 1^. sin. 9. cos. <fi — sin. 6. cos. if/, sin. *ip), — ■y.rf2.(sin. «. cos. e. cos. 4- sin. (p. cos. <p-f sin. 6. sin, 4-. sin.'^); PART I.— BOOK I. 137 c'. (sin. ^. sin. ^+cos. 6. cos. x)/. cos. <p)+c^''. (cos. \f/. sin. p — cos. 8. sin. 4'. COS. <p) J '^^■^'"'^^'"•—^"''-^■!^^ = —c. sin. 9. sin. <p. +c'.(sin. 4-. COS. (p-^cos. S. cos. i)/. sin. <?) +c". cos. ^'^ COS. 9 + cos. 9. sin. ^. sin. (p). If we deteiinine 4^ and 6, so that we may have sin. 9. sin. i|/ c" — c' , . = ,^ — 1=7=^ ; sin. 9. cos. J/ = ~>~; ,„ — , which gives c COS. 9 = / „ ., .,, we shall have * 3c,.dii . — y„,'dx , , . 2WJ. dt '■ = \/c*^c'*-^c"* + •/ subducting x,iidyi,, from y„,.t/x,„, and making the terms whose variable parts are the same coalese, we obtain ^m-dyn, — .y/„.rfa;,„ = (x.rfy — y.dx). cos. «-{- (xdz — zdx), sin. #. COS. i//-|-(^.t?2 — ^z.dly), sin. 6. sin. •vf/; and substituting for x.dy — y.dx, s.dz — z.dx, &c. their values d ,c" ,, we obtain c. cos. i d. sin. «. cos. -^/^-^fd'. sin. ^. sin. 4'; =x„,.f^y,„ — ViiA^iiii ^y ^ similar analysis we arrive at the expressions for Xm-dz^,! — z^^-dx^^,, ym-dz^i, — ~iii-^y,i,> which are given in the text. c"« J. c" ' * For sin. ' 6. sin.* i^ -f sin. * 6. cos. * i|/= sin. - « = vTv a j.'X'a' **' ''°^' ' ^'~^ — *'"' * * c^+d^+d" f For substituting in place of cos. 6^, sin. 6. cos. xj',. sin. S. sin. •vJ/„ these values, we shall have rf< := V c*4-c'--|-c'S and if we substitute for c, c', c", their values, ■\/c^-J-c'^-f c*", cos. «, — 'V^c*+c'*-j-c"*, sin. «. cos. 4". +v^c2+'c*-f-"c», sin. «. sin. ■!•, the expression "•"'• ill ~" will become v6'-\-c^-{-c', (sin. 6. cos.S. cos.i?, — sm.^.sm.if. ISS CELESTIAL MECHANICS, ^^^^ x„,dz„r-z,Ar,„ ^ 0 ; ^.ILid^H^j:^, = 0 ; dt dt ,'. the values of c' and c" vanish with respect to the plane of x^,, and ^„,, determined in this manner. There exists only one plane, which pos- sesses this property, for supposing that it is the plane of x and i/, we shall have Z;«. "'-^-^--^-•^"- = c. Sin. 9. COS. , ; Im. y'-d^''^-'-^^'" = dt _ dt — c. sin. 9. sin. ip ; If these two functions are put equal to nothing, we shall have sin. 9=0, which shews that the plane of a-,,, and y,,,, then coincides with V dii 1! • dx the plane of x and y. Since the value of "Zm. ' '"' " fji "' — ~ is equal to \/c*+c* + c"*, whatever may be the plane of x and y, it follows that the quantity c* + c'*+c"« is the same, whatever this plane may be, and that the plane of x„ and y,„, determined by the preceding , , , p . X dy — y .dx analysis is that, with respect to which the function s/h. '" '^"' " — - is a maximum ; therefore, this plane * possesses these remarkable pro- cos. ■\. sin. (J) — sin. i. cos. ^. cos. --i^. cos. <?)-l-sin. i. sin. -^ -f cos. ■^. sin. (p — sin. (i. cos. C. sin. '\f/. cos. ip) = ^c^-t-c^'-f-c"'', (sin. «. cos. d. cos. $ — sin. i. cos. S. cos. if) = 0 the same is true respecting the expression 2.jb. -= — : . * As the cosines of the angles which the axes of z„, maizes with the axes z, y, x, i, e. the cosines of the angles which the plane x^„. ?/,„, makes with the planes x, y; x,z;y,z, (see note to page 1 33, ) ai-e equal to cos. 6, sin. (t. cos. ■4', sin. 6. sin. ■^, it follows that when we have the projections c, c', c", of any area on three coordinate planes, we have its pro- jection 2M.(a:„,.(/y,„ — y,i,-^^ii^ °n ^^^ plane a;„/y,„ whose position, with respect to the three planes x,y ; x^ ; y,z, is given. In like manner it follows from the exression, 2.n!. -{ -J'JlJjiJ y'" ' ^" V , which has been given in the text, that for all planes equally inclined to the plane on which the projection is the greatest, the values of the projection of the area are equal, for supposing the plane of x, y to be the invariable plane, then ( xdu — y.dx (.,,,, ., , (■ xdz — z.dx \ 2.m. ■{ — /t~~' f ' '"''" ^^ *"^ greatest possible, 2.m. I j- , PART L— BOOK I. 139 perties — first, that the sum of the areas traced by the projection of the radii vectores of the respective bodies, and multiplied by their masses, is the greatest possible ; secondly, that the same sum, vanishes relative to any plane, which is perpendicular to it, because the angle ip is unde- termined. By means of these properties, we shall be able to find this plane at any instant, whatever variations may be induced in the respective positions of the bodies by their mutual action ; we can, in like manner very easily T 2 2.7n. -J ~ — T — \ , are respectively equal to nothing, V 2.m. < '' ' — > --^ >. COS. 6, dt j Since c, c', c", are constant quantities, and proportional to the cosines of the angles which the plane on which the projection of the area is a maximum, makes with the coordinatfe planes, it follows, that the position of this plane is always fixed and invariahle ; and as the quantities c, c', c", depend on the coordinates of the bodies at any instant, and on the velocities dx -J—, &c. parallel to the coordinates, when these quantities are given, we can determine the position of this invariable plane ; we have termed this plane invariable, because it depends on the quantities c, c', c'', which are constant during the motion of the system, provided that the bodies composing it are only subjected to tliis mutual action, and to the action of forces directed towards a fixed point. (See page, 128.) Since the plane ^, ^ is indetermined in the text, we conclude, that the sum of the squares of the projections of any area, existing in the invariable plane, on any three coordinate planes passing through the same point in space is constant ; consequently,, if we take on the axes to any coordinate planes y, z; x,z; x, y, lines proportional to c, c', cu, then the diagonal of a parallepiped, whose sides are proportional to those lines, will represent the quantity and direction of the greatest moment, and this direction is the same whatever three coordinate planes be assumed, but the position in absolute space is undetermined, for the projections on all parallel planes are evidently the same. The conclusions to wliich we have arrived, respecting the projections of areas on coordinate planes, are in like manner applicable to the projections of moments, since as has been observed in Note, page 28, these moments are geometrically exhibited by triangles of which the bases represent the projected force, the altitudes being equal to perpendiculars let fall from the point to which the moments are referred, on the direction of the bases. When the forces applied to the different points of the system have an unique resultant, V; then smce the sum of the moments of any forces pr<jjected on a plane is equal to the moment of the projection of their resultant, it follows necessarily, that 140 CELESTIAL MECHANICS, find at all times the position of the centre of gravity of the system, and for this reason it is as natural to refer the position of the coordinates x and y to this plane, as to refer their origin to the centre of gravity.* 22. The principles of the preservation of living forces, and of areas, obtain when the origin of the coordinates has a uniform rectilinear motion in space. To demonstrate this, let X, Y, Z, represent the co- ordinates of this origin, supposed to be in motion, with reference to a fixed point, and let us suppose X = X-\-x;, y = Y-\-y,\ z = Z-\-z;y x' = X+x;;y'^ Y+y/, z = Z+z/, &c. j\, 7/,, z^ ; x^, &c. will be the coordinates of m, ?«', &c. relative to the the unique resultant V and the point to which the moments are referred, must exist in the invariable plane ; *.* the axis "of this plane must be perpendicular to this resultant, and as —p-, -p, — , are equal to the cosines of the angles which F makes with the coordinates, and as c c' c" sJc--^d^-\-c"'^' x^c-' + c^+c"^ ' ^c^ +c'» + c"', are equal to the cosines of the angles which the axes to the invariable plane makes witii the same coordinates, we have cP+c'Q+c//R ^-^^ =: 0, V cP+c'Q+c^R ~ 0. (See note to No. 1, page 7.) \/c^-\-c'^+c'^' * Besides the advantages adverted to in the text, it may be observed, that our inves- tigations are considerably simplified by the circumstance of two of the constant arbitrary quantities c, c, c', vanishing when we make the plane of projection the invariable plane. It may also be remarked that this plane always subsists when the bodies composing the system are not solicited by any forces beside those of mutual attraction, and of forces directed towards fixed points ; nor is the position of this plane affected in any respect when two or any number of bodies impinge on each other ; for as we have before ob- served, these impacts dont cause any change in the expressions Py — Qx, &c. — on the equality of which to nothing depends, the principle of the conservation of areas, and the PART I.— BOOK I. 141 moveable origin. We shall have by hypothesis, d*X= 0; f/*F= 0; d*Z= 0; but we have by the nature of the centre of gravity, when the system is free O = z»j.(c?*Z+ d*x,)—^m.P.dt* ■* 0 = zm.(rf» Y-\-d*y}~-^m.Q-dP ; 0=-s,)n.(d'-Z+d^z,)—^m.R.dt'; position of tliis invariable plane. The practical rule for the determination of this plane is gi^'en in the exposition, Du Systeme du Monde, page 207, the investigation of this rule will be given in No. 62, of the second book. We shall see in No. 26, chapter 7, that the consideration of this plane is of great service in the determinations of the motions of a body of any figure whatever. * 0 = 'S..m.d^x—lM.P.dt^ ; o = '2..m.d^ij—-S..m.Q_.dt- ; 0 = ^.m.dz^—'S.m.R.dt' ; substituting in place of d'x, d\i/, d'z, we obtain the expression in the text; and since d^X is by hypothesis equal to 0, tlie expressioi\ 0 i: ^.m.(d^x,+d^X — ^.P.dt"^) = 2.w. rf*i,-}-rf»X. 2.JB. — ^.m.P.dt ^ = 'S.m.dx,^ — X.m.Pdf,-, &c. ; in like manner, substituting for ix, 3_y, &c. in the equation (PJ, we obtain 0 = 2.m. I SX+Jx, } -^ _ p.j +2.,«.(?y+ 5^, I g— Q.j + &c. = but as by the nature of the centre of gravity 2. m. < -j-^ f ' ^•'"' i "^T ^' i ' ^^' ^^ respectively equal to noo thing, and also d'x = rf*j, d-ij — d'-t/, &c. the preceding expression becomes 0 = 2...S. . { £^ _P. ] + 2.n..3,, { 4;f -Q. } &c. UQ CELESTIAL MECHANICS, and by substituting SX+Sx^, jY+S^i/,, SZ+Sz^, &c. in place of J'.r, ii/, fz, &c. the equation (P) of No. 18, will also become . \ t* -«. -w-\ f + xmJz. i—r-^ — Ry-, ' ( dt* r which is precisely of the same same form as the equation (P), if the forces P, Q, R, P\ depend only on the coordinates ^^, 3/^, z^, x', he. Therefore by applying to it the preceding analysis, we can obtain the principle of the preservation of living forces and of areas, relative to the moveable origin of the coordinates. If the system is not acted on by any extraneous forces, its centre of gravity will move uniformly in a rectilinear direction in space as we have seen in No. 20 ; therefore, by fixing the origin of the coordinates X, y and % at this centre these principles will always have place, X, F, and Z, being in this case the coordinates of the centre of gravity, by the nature of this point, we shall have 0 = 's-.mx ; O = 'S.m.y ; O = 'zm.z $ consequently we have ( di ) dt dt * S.jn. \ dt ■2.m.XJY+^mx.dY-\-'Z.m.Xd.y,~^-2.m.x^dy^ dt l.m.Y.dX — "Z.m.yAX — Im.YJx — 'S.My.dx, , ^ „ ^ j - j — ■^' ■^' ' , and as ^.mx,, ^.my/, ^m.dx,, ^.m.dy^ are respectively equal to nothing by the nature of the centre of gravity, the preceding PART I.— BOOK I. 143 2.WJ. ^- = 3- Z.m. dt* df- +2.WI. dx;-\-dy'+(h\' 7 „ dt y thus the quantities which result from the preceding principles are com- expression becomes equal to rff ^ rf« * 2.m.dx^ =-. 2.m.rfX' + 22.n!^x,.rfX+ 2.m.dx,^, enxdas^dX. ^.m.dx, = 0, we have 2.m.rfj;' _ rfX*. Z.m+^.in.dx*, ;• dx^+dv-'+dz^ dX'+dY^JUdZ^ ^ , „ <&/ +rfy »+(fc/- . „^ 2.W. f^z — = \nx ■ " — > 2.m + 2.jn. — 1— i--^;~I— i_ &c=c+2?. If all the bodies were concentrated in their common centre of gravity, X/, i/j ; dx^, dyj ; would vanish, therefore the second part of the first members of the preceding equation would .^ J ,,, X.dY—Y.dX dX'- + dY' +dZ^- vanish, and we would have 2.»». = c, -j-^ '2m~c-^-2<p. Consequently, it appears from what has been established in this number, that when the bodies composing the system are not acted on by foreign forces, the quantities which are concerned in the principles of living forces, and of areas are composed of quantities which would have existed, if all the bodies of the system were concentrated in their centre of gravity ; and 2dly, of quantities which would obtain if the centre of gravity quiesced, the former description of quantities are represented respectively by -^ 2.w, XdY^YdX , , , , dx,^ +di/,^- +dz,^ xdy—y,.dx,, , 2m — . = , and the latter by S/n ^ T J/ t / j.„ «, y y< "-^z!. jj^g dt ■' af" ' dt first indicates what obtains in consequence of the progressive motion of the system, the second what arises fi-om a rotatory motion, about an axis passing tlirough the centre of gra- vity. (See No. 25.) If the origin of the coordinates x,y, z,be transferred to a point of which the coordinates are A, B, C, the expression for the projection of area on the plane x, y, becomes ^^_ {x-A)dy^y-B).dx ^^^(^J>r:y±\ _ A.^m.dy+B.Zmdx ^^^^ ^ dt \ dt § dt ' v J t-™ J jxr^ jxr „ AY.mdy-\-Bzm.dx , A.dY4-B.dX „ 2m. dy, Zm. dx=d Y 2m, dX. 2m ; — -^ becomes — -^ 2.)?j. 144 CELESTIAL MECHANICS, posed, 1st of quantities which would obtain, if all the bodies of the system were concentrated in the centre of gravity ; 2dly, of quantities relative to the centre of gravity supposed immoveable ; and since the first described quantities are constant, we may perceive the reason why the principles in question have place with respect to the centre of gravity. Therefore if we place the origin of the coordinates at this point, the equation Z, of the preceding number will always subsist ; I'rom which it follows that the plane which constantly passes through x.dy — ij,dx this centre, and with respcet to which the function S.ot. < - dt .'. the projection of the area on the plane x, y, with respect to the new origin becomes ,^ . B. dX—A.dY, , . . equal to c-j . Sw;, and similar expressions may be derived for the pro- jections on the planes x,z,y,z, From tliis it appears, that for aJl points in which B.dX—A.dY ."Lin = 0 the value of c will remain constantly the same, but it is evident that this equation will be satisfied, if the locus of the origin of the coordinates be either the right line described by the centre of gravity, or any line parallel to this line, consequently for all such lines the position of the invariable plane will remain constantly parallel to itself; however, though for all points of the same parallel the position of the invarialile plane is constant, yet in the transit from one parallel toanother the direction of this plane changes. If the forces which act on the several points of the system are reducible to an unique resultant, by making the origin of the coordinates any point in this resultant, the quantities c,c',c", and therefore the plane with respect to which the projection of the areas is a maximum, will vanish, if the locus of the origin of the coordinates bo a line parallel to this resultant, the value of the projection of the area with respect to this line on the plane ar,_y, will be constant and equal to — '■ . -^m for c in this case vanishes, if the locus of the origin of the coordinates be a right line diverging from this resultant, the expression BdX—AdY -J. • £»2 IS susceptible of perpetual increase. From these observations it appears that when the forces admit an unique resultant, that point with respect to which the value of x/ c* + 1' - c" - is least of all is a point so circumstanced, that the axis or perpendicular to the plane of greatest projection passing through this point, is parallel to the direction of the unique resultant ; PART I.— BOOK I. H5 is a maximum, remains always parallel to itself, during the motion of the system, and that the same function relative to every other plane which is perpendicular to it, is equal to nothing. The principles of the conservation of areas, and of living forces, may be reduced to certain relations between the coordinates of the mutual distances of the bodies composing the system. In fact, the origin of the coordinates r, j/, z, being supposed always to be at the centre of gravity; the equations (Z) of the preceding number, may be made to assume the following form (■ (It \ C dt S' c".^.m = ^.mm'. S^y-y)'d^''-dz)j^i^'-zUdy'-^dy)}^^^ It may be remarked, that the second members of these equations u * This expression is proved to be true with respect to three bodies in the following man- ner and as the same reasoning is applicable to any number of bodies whatever, it may be considered as a general proof ^ / <si^—A {di/—d,'l)—{y'—n). (dx'—dx)-) ,,dy , dy' , dy C dt ) dt dt dt I dy , J dx' , , dx\ .da/ . dx , „ „ di/' J^ mm . X ■jr—mmif. y- -f ramV. — + wm'.v-; mm'y.-rr -{-mm" .xf' -^^ dt dt ' ^ at ^ dt ^ di ' dt ~fnm".x.'^jf.-mm".x".^+n,m".x^-mmf'.i/'. ^+ mm".y".^ +mm".y.— -mmf'.y J + m".m'.x'. Jl -m".m\x' %- -m"m'x" % + m'm".x'M- "t dt dt dt dt «.•'„, , // dx" , „ , ji dx! , „ , . dx" , ., . dx' — w m.y .--. +m"nf.y". -T-+m"fn'y'. m'm'.y . — dt " dt ^ dt ^ dt U6 CELESTIAL MECHANICS, multiplied by dt, express the sum of the projections of the elementary areas, traced by each line which joins the two bodies of the system, of which one is supposed to move round the other, considered as im- moveable, each area being multiplied by the product of the two masses, which are connected by the right line. and by concinnating it comes out equal to , (x'di/ — i/.dx' ) , , ( xdy — y.dx~> , Cx'.dij — y'.dx'l ""• j-^V— } + "'"• {t^ 5-"""- I d/ } \ dt S ^ I dt 5/^ L tit i sad as in the case of three bodies „ ( x". dy"—i/'- da!' 1 ^ / . , . //^ ^ f =^dy—y.dx \ _|.m". J dt \ ■'■ '^' ^"'—'' (»»+»»+>») ~"^ { — ^^ \ + ^V ^'::f^t^}+m".^ ^^jMlr-^^J^-^ + m,n' {^^ ^ ^, ix".dy"-y".dx" \^ , „ {J>'.df-f.di!' \ + «"^' \—ht — ) + """ { — Jt — r By the nature of the centre ofgravity we have w«4- >"''»''+"'" *"=0 2nd also mdy A- m' dtf ^td'.dy" — 0 /. their product vanishes z, e, m''x'Jy+m'^x'di/-\-m."-x".dy"-\-mm'x.dy +mm". x". (fy-j- mm'^di/ ■\-m'm".a!'dy'-i-m.m",xdi/'+m"m'.x'.dy"=0 .'. we have m » x3y +m''t^ulj/-\-m''.'-a!'.dy''s=—mm'.x'dy—mm''.x''.dy—mm'.xdy'—m'm"'.x''dy—mmf'j:dif''. — m'.m'^di/', and by multiplying my+m'i/+m"i/', into mdx-\-m'dj/-\-ni"dx"; — ot' ydx PART I.— BOOK I. 147 By applying to the preceding equations, the analysis of No. 21, it will appear, that the plane passing constantly through any of the bo- dies of the system, and with respect to which the function ( dt 5 is a maximum, remains always parallel to itself, during the motion of the system, and that this plane is parallel to the plane passing through _fn" i/.dx'—m"^7/'. dx" = + mm'.y'. dx + mm" i/'. dx + mm'y.d^ + m'm"y"dx! + mml'.ydx!' ■\-m".m'.y'Jx", .•. adding these quantities together we obtain L dt i l dt S ' X dt i .-. if in the expression for c{m+m'+ m") we substitute in place of the sum of the functions whicli are multiplied by the squares of the masses, the quantities wliich are equivalent to theni we shall obtain c {m + m' -f m")= - nun' \^:i!jii±^ I _ „„„" i f:^^/:^! _ „„,, r ..^^^^ 7 (. dt } I dt y \ dt s which is equal to to the expression which has been given above for the value of Im.rh': i JX'—x) d,/ —d, /)—,/— y (dx'—dx \ ■ I It ) 148 CELESTIAL MECHANICS, the centre of gravity, and relatively to which, the function s.ffj. ^ ' ■ — '- is a maximum. It vpill also appear that the se- cond members of the preceding equations vanish with respect to all planes passing through the same body, and perpendicular to the plane in question. t The equation (Q) of No. 19, can be made to assume the form* I dt •z.fmrd. Fdf; this equation respects solely the coordinates of the mu- * When there are but three bodies S.w.rfx ^=:»iix* + »2'<ir'* + »n".(fji;", * but by the nature of the centre of gravity we have mdx-^m' dx' -\-m" .dx" —0 and .•. m^dx- -\-m'.^ dx'^ + ?n".^ rfx"' + 2m.m'. dx.dx'-\-2mm". dx.dx" + 2m'm". dx'.dx", = 0, and multiplying l^.m.dx- by 2w. we obtain, m^ t/x^-j-m'.^ dx'^ -f m"r dx"- + mm' dx'- + m'm" . dx'^ + m'm.dx^+m"in dx''+}nm".dx"^ +m'm".dx",' if we substract the previous equation from this we get, mm'.dx"' +m.'m".dx^ +m'm.dx'^ -\-m".tn.dx^ +mm"Jx".- ■¥m'm".dx"' —2mm'. dx dJ— <2.mm'. dx.dx! '—%n.m''.dx'.dx«~ mm'. [dx'—dxY +tn'm" {dx"—dj/y + mm"{dx'/ — dx)- — 2.)?2jn'. (rfx' — dx)' = 2.)». {2m. (dx^), and in like manner we derive 'S.nim' [dy — dy) ' = 2)h. (2wi. (/y^), 2.»nm'. {dz — dz) ^ = 2wi. (2j?z. dz'), .-. we have , ^ dx'—dxY +idy'—dy)^ -\-{dz'—dz'^ \ ^ ,^ ,_ ^ /. d r , ^ , SwiTO -} A , j- = c. 2m + 2.jn. (2. ^m.fm. \P.dx-\-Qdx -j- Qfl^) = const. — 2 2m. l.fmmfdf, (substituting — 2/nw('.y(i/'in place of "S-./m (Pdx^Q.dy-\-ndz). (See No. 19, page 113.) ■f When the origin of the coordinates is in the centre of gravity of the system, the quan- tities c c' c ', are constant and .•. the position of the plane, with respect to which the function 5)n. ■} — 5L_jZJ — V is a maximum, remains the same during the motion of the system, .•. as the quantity 2)«, would occur both in the numerator and denominator of the expression for the cosines of the angles which the plane with respect to which the function S.m.in' } J LLJl J-T^j-HSIA L > is a maximum, makes with the three coordinate planes, it is evident that the values of the angles which the invariable plane makes with three coordinate planes, is the same in both cases, from these considerations it appears that the invariable plane may be determined at each instant by means of the relative velocities of the system, without a knowledge of their fliio/if/e velocities in space. (See Notes to page 139.) PART L— BOOK I. 149 tual distances of the bodies, in which the first member expresses tlie sum of the squares of the relative velocities of the system about each other, considering them two by two, and supposing at the same time that one of them is immoveable, each square being multiplied by th<> product of the two masses which are considered. 23. If we resume the equation {R) of No. 19, and differentiate it with respect to the characteristic <?, we shall have l.m V. iv=2m. (P.Sx + QJy + RSz) ; and the equation (P) of No. 18, will then become 0='Z.m. \sx. d.-^-tSjf. d. -^ +<?^.c?. -^\~-L.m.dt.vSv. i. dt dt dt 3 Denoting by ds^ ds' &c. the elements of the curves described by m, m &c. ; we shall have vdtzids; v'dt=ds';kc. ds = ^dx^ + di/* + dz'; &c. from which we can obtain, by following the same process as in the analysis of No. 8, 2.mJ. (_vds) = ^.m. d. (JlLt±^Ml^Ll^±JL\ . By integrating with respect to the differential characteristic rf, and making the integrals extend to the entire curves described by the bodies »2/k', &c. we shall have J..S.fmvds = const. + 2.7». ^^-S^+dy.Sy + dz.Sz.\ in this equation the variations Sx,Sy,Sz, &c and also that part of its second member, which is constant, refer to the extreme points of the curves described by m,m', &c. ISO CELESTIAL MECHANICS, From which it appears that when these points are invariablcj, we shall have 0 = iJ.fmvds ;* which indicates that the function 'L.fmvds is a minimum. It is in this, that the principle of the least action, in the motion of a system of bodies, consists ; a principle, which, as we have seen, is only a mathema- tical result of the primitive laws of the equilibrium and motion of bodies. It is also apparent, that this principle combined with the principle of living forces, gives tlie equation (P) of No. 18, which contains all that is ne- cessary for the the determination of the motions oi the system. Finally, it appears from No. 22, that this principle obtains, even when the origin of the coordinates is in motion ; provided that the motion is uniform, its direction rectilinear and the system entirely free.t * By substituting for ds, ds their values v.dt^ v'dt^ the expression "S-./wds will become 2.y»!D.'«ft, and as ymv-'dt is the sum of the living forces of the body m during the motion; t.J~mv.^dt vi'ill express the sum of the living forces of all the bodies of the system during the same time ; therefore the principle of the least action, in fact indicates, that the sum of the living forces of the system, during its transit from one given position to another, is a minimum, and when the bodies are not actuated by any accelerating forces, the velocities v, v, and the sum of the living forces at each instant, are constant, (see No. 18, page 1 1^. ).■. 2. fmv.'^dt=z'2mv, 'J'dt, and the sum of the living forces for any inteiTal of time is proportional to this time, consequently in this case the system passes from one position to another in the shortest time. Since therefore the expression Yfv.dis is the same as I.Jmv^dt La Grange proposed to alter the denomination of the principle of least action, and to term it the principle of the greatest or least living force, for by contemplating in this manner, it is equally appli- cable to th(^ states of equilibrium and motion, since it has been demonstrated in the notes page 119, that incase of equilibrium the vis viva is either a maximum or a minimum ; from what precedes ic appears that, as La Place observes in his Systeme du Monde, the true economy of nature is that of tl;e living force, and it is this economy which we should always have in view in the construction of machines, which are always more perfect according as less living force is~c6nsumed in producing a given effect'. \ W ith respect to the extent of the different principles which are treated of in this fifth chapter, it is important to remark, that the principles of the conservation of the motion of the centre of gravity, and of the constrvation of areas subsist, even when by the mutual action of the bodies, they iindergo sudden changes in their motions, which renders these ■jrii ciples extremely useful in several circumstances, but the principles of the conservation PART I.—BOOK I. 151 of the vis viva, and of the least action, require that the variations of the motion of the system, be made by imperceptible gradations. The principle of the least action differs from the other principles in this, that the other principles are the real integrals of the differential equations of the motion of bodies, whereas this of the least action is only a singular combination of these equations, in fact it being established that 'S./mv.ds is a minimum by seeking by the known rules, the conditions which render it such, and making use of the general equation of the conservation of living forces, we should find all the equations which are necessary to determine the motion of each body. The principle established in this number was first assumed as a n>etaphysical truth, and was applied by Maupertius to the discovery of the laws of reflection and refraction, however it ought not to be deemed ajinal cause, for we can infer analogous results from all relations mathematically possible between the force and the velocity, provided that we substitute in this principle, in place of the velocity, that function of the velocity by which the force is expressed, (see next chapter, page 154,) and so far from having been the origin of the laws of motion, it has not even contributed to their discovery, without which we should be still debating what was to be understood by the least action of nature. 152 CELESTIAL MECHANICS, CHAPTER VI. Of the laws of motion of a system of bodies, in all the relations mathematically possible between the force and the velocity. 24. It has been already remarked in No. 5, that there are an infinite number of ways of expressing the relation between force and velocity, which do not imply a contradiction. The simplest of all these relations is that of the force proportional to the velocity, which as we have ob- served, is the law of nature. It is from this law that we have derived, in the preceding chapter, the differential equations of the motion of a system of bodies ; but it is easy to apply the same analysis, to all relations mathematically possible, which may exist, between the force and the velocity, and thus to exhibit under a new point of view the general prin. ciples of motion. For this purpose, let F represent the force and v the velocity, we have F zz <!> (y^ ; (p (y) being any function whatever of v ; let <p' (w) denote, the difference of <p(y) divided by dv. The denominations of the preceding numbers always remaining, the body tn will be solicited parallel to the axis of s by the force <? (vj. — - . * diT \ dx '} In the following instant, this force will become (p (v). -jz"^ '^•f f C"^)' — r r * ds being the differential of the line described by the body, the cosine of the angle dx which the direction of the motion makes with the axis of x is equal to -p, .-. the force F ds dx or p (v) resolved in the direction of the axis of j ss <p (f).-?- • or PART L— BOOK I. 153 ^M. $ +d. C-^ . ^ Y because ^ = f • Moreover, P, Q, i?, ^ ds \ V dt / dt being the forces which solicit the body m parallel to the axes of the co- ordinates ; the system will, by No. 18, be in equilibrio in consequence of these forces, and of the differentials, {^dx <w)? \dy <p{v)\ , $ c?s (fip)\ O. •<-; — . f , a. \ — ;- • f , Urn < —r- • t , \dt f J I dt V S Xdl V S taken with a contrary sign ; therefore in place of the equation (P) of the same number we shall have the following : 0 = ^,m. \ ix. d. Y-1.M^ Pdt I + Sy. d, S^y^-:^ i (dt V i -^ Idt V - Q.dtl + Sz. dA^.-^^Rdtl; (S) which only differs from it in this respect,that— > — > — f are multiplied by dt dt dt the function A_i, which in the case of the force proportional to the velocity, may be assumed equal to unity. However this difference renders the solution of the problems of mechanics very difficult. Notwithstand- ing, we can obtain from the equation (S), principles analogous to those of the conservation of living forces, of areas, and of the centre of gravity. By changing Iv into dx, Sy into dy, Sz into dz, &c., we shall have 2.J«. V. dv. dt. <p' (y) ; * X * Substituting ds in place of v.dt, the expression 154 CELESTIAL MECHANICS, and consequently X.fmv.dv.(p'[v) = const. -{-l.fm. (^P.dx+ Q.dy + R.dz), If we suppose that I.m.(P.dx-i-Q.dt/ + R.dz) is an exact differential equal to d\ we shall have i..Jmv.dv.(p' (t>) = const. + x j (T) which equation is analogous to the equation (i?) of No. 19, into which it is changed in the case of the law of nature, or of 9' (tr)zzl . Therefore, the principle of the conservation of living forces obtains in all laws mathe- matically possible between force and velocity, provided that we under- stand by the living force of a body, the product of its mass by double the integral of its velocity, multiplied by the differential of the function of the velocity which expresses the force. If in the equation (5), we make Sx'—Sx-^Sxl, Sy' = Sy-\-Syl, i^= Sz + iz', Sx" = Sx+^x",j &c. we shall have by putting the coefficients of Sx, iy, Sz, respectively equal to nothing becomes 2.«. {^. rf. { ^. <p (V) J +dy. d. {|. <p (v) } + dz. d. {J . <p{v) } } and by taking the differential it becomes. -•«• \ ds 1.?(«)-2.«.| J-p: ]-d^s4{v)+ Z.m. J ^ -^ ^ \ . d. (p (v)~ 2.m. d» s. f («) —2. m. d^s. ^ (v) ^"Z.m.ds. d, (p (v) and this last quantity is equal by substitution to 2,m, v>dtdv. f ' (v). PART I.— BOOK I. 155 These three equations are analogous to those of No. SO, from which we have inferred, the conservation of the motion of the centre of gravity, in the case of nature, when the system is not subjected to any forces but those of the mutual action and attraction of the bodies of the system. In this case l.m. P, ^.m. Q, "Z.m. R, vanish, and we have dx a(v) . „ dy a(v) const. = l.m. -—. -^-i ; const. = 2w«. -^. ^-^^-^ ; dt V dt V _ dz <p(v) dx ((i(v) . / \ da: const. z= 2w . I^^jjw.— -. ^^^-^ IS =:ot. fflfiy). -^' dt V dt V ^^ '' ds and this last quantity is the finite force of the body, resolved parallel to to the axis of x ; the force of a body being the product of its mass by the function of the velocity which expresses the force. Therefore in this case the sum of the finite forces of the system, resolved parallel to any axis, is constant whatever may be the relation between the force and the velocity, and what distinguishes the state of motion from that of repose, is, that in this last state, the same sum vanishes. These results are common to all laws mathematically possible between the force and the velocity ; but it is only in the law of nature, that the centre of gravity moves with an uniform motion in a rectilinear direction. * Again let us make in the equation (iS) Sx'=^— + 9x:iix"='^— + Sx" &c. y V . xdx , . . , x'.Sx » / a " if the variation Sx will disappear from the variations of the mutual distances X 2 • It is evident that the centre of gravity does not move uniformly in a right line when P, Q, R, vanish, except when -^ is equal to unity, for it is only in this case that we could prove from the expression, const. —2.W. -J-. -^ , that dX the differential of the co- ordinate of the centre of gravity is constant. 156 CELESTIAL MECHANICS, f,f, &c. of the bodies composing the system, and of the forces which depend on these quantities. If the system is not affected by extraneous obstacles, we shall have, by putting the coefficient of ix equal to nothing o=..„,.|..,(^.^)_,...(^.Ka)| + "s-m.^Py — Q.x)dt, from which we deduce by integrating, we shall have in like manner i \ at / v d' = z.^. {^^^=^y ?^+ z> {Qz^Ry).dti c, c', c", being constant arbitrary quantities. If the system is only subjected to the mutual action of its component parts, we have, by No. 21, Im. [Py — Qx) = 0 ; s»z. [Pz^-Rx) = O sw. CQz — Ry)=0; also m] x -^ — y. —i.'^^ is the moment of C dt dti V the finite force by which the body is actuated, resolved parallel to the plane of x and y^ which tends to make the system turn about the axis of z J therefore the finite integral s.w. J-^fc^^ LfM is equal to the sum of the moments of all the finite forces of the bodies of the system * Tlie integral of this expression is equal to 2.ot i x. -j- .^^ — /dx. (-^. --^^' \ dx <p{v) ( dy.dx (p(u) -v 1 _ _ xdy—ydx ip(«) '"^* dt- V +-^{,~dt — 7yj ~^-'"- — It — •~' PART I.— BOOK L 157 to make it revolve round the same axis ; consequently this sum is con- stant. It vanishes in the case of equilibrium ; therefore, there is the same difference between these two states as there is relatively to the sum of the forces parallel to any axis. In the law of nature, this property indicates that the sum of the areas described about a fixed point, by the projections of the radii vectorcs of the bodies is constant in a given time, but this constancy of the areas described does not obtain in any other law.* By differentiating with respect to the characteristic S, the function ^.Jhi. (p {v). ds ; we shall obtain S.'z.fm.(p(y')ds ■=. 'z.fm.(p(y').S.ds-{-t.fm.Sv.(pXv).dS'y but we have sa,^d.Jd.+dvMy+dzMz l_ W_r^ cl^ dz ^^ i ds V idt dt -^ dt S therefore by partial integration we shall obtain *^ r f \ r T. fnqiQv) ^dx . . dy . , dz , 1 V idt dt dt 3 c ^dt V ' ^ ^dt V ' ^dt V ' S ■\-^.JmJv.<p'{v).ds. The extreme points of the curves described by the bodies of the system * As the factor is variable in every other case beside that of nature, it follows that though the quantity 2.m. -I '-^^^ — \ -—^ is constant and equal to c, still that part of it cxdy — y.dx 7 . 2.W. i ~j^f — f >s not constant. 15S CELESTIAL MECHANICS, being supposed fixed, the term which is not affected by the sign/ must disappear in this equation; therefore we shall have in consequence of the equation (S), i. z.fm.(f{v).ds = x.fm.Sv.(p'{v).ds—^.fmdt{FSx + Q.Sy + R.iz) but the equation (T) differentiated with respect to S gives ^.Jm.Sv.<f'(v).ds=^.fmdt{PSx-\-Q.Sy-i-R.Sz) ; therefore we have OzzS.'Z.fm.<p(y).ds. This equation corresponds to the principle of the least action in the law of nature. vi.(p(v) is the entire force of the body m, thus the prin- ciple comes to this, that the sum of the integrals of the finite forces of the bodies of the system, respectively multipHed by the elements of their directions, is a minimum, presented in this manner, it answers to all laws mathematically possible between the force and velocity. In the state of equilibrium the sum of the forces multiplied by the elements of their directions vanishes, in consequence of the principle of virtual velocities, what therefore in this respect distinguishes the state of equilibrium, from that of motion is that the same differential function, which in the state of equilibrium vanishes, gives in a state of motion by its integration a minimum. PART I.~BOOK I. 159 CHAPTER VII. Of the motions of a solid body of any figure whatever. 25. The differential equations of the motions of translation and rotation of a solid body, may be easily deduced from those which have been given in the fifth chapter; but from their importance in the theory of the system of the world we are induced to develope them in detail. Let us suppose a solid body of which all the parts are solicited by any forces whatever. Let x, y, z, represent the orthogonal coordinates of its centre of gravity, and let x-\- x',y-\-y\ s+;s', be the coordinates of any molecule dm of the body, then *',y, s', will be the coordinates of this molecule with respect to the centre of gravity of the body. Moreover, let P, Q, i2, be the forces which solicit the molecule parallel to the axes of X, of y, and of ^,. The forces destroyed at each instant in the molecule, parallel to these axes, will be by No. 1 8, -S^^pLl^dmArF.dUdm', ^Y^y^^^\dm\Ci,dUdm', — \ — ^.dm-\-R.dt.dm ; (the element dt of the time being considered as coustant.) 160 CELESTIAL MECHANICS, Therefore it follows that all the molecules actuated by similar forces should mutually constitute an equilibrium. We have seen in No. 15, that for this purpose, it is necessary that the sura of the forces parallel to the same axes, should vanish which gives the three following equations C dt» i ' sY'y+f^'\.dm=S.Qdm; S.l±5pl£j^.dm^S.Rdm, the letter S being here a sign of integration relative to the moelcule dm, which we should extend to the entire mass of the body. The variables' x,i/, z, are the same for all the molecules, therefore we can bring them from under the sign S ; thus, denoting the mass of the body by m, we shall have o d^x J d'^x c- d^if , c?*w „ d*z , d*z S. . dm=7n. __; S. —d, dm—m. -JLi S. ■— . dm-=.m -' dt^ dt' dt* dt^' dt* dt^ Moreover by the nature of the centre of gravity, we have, S.x'.dm = 0 ; S.j/.dm — 0 ; S^.dm = o * therefore S, ~.dm-0 ; S. ^.dm=0 ; S. ^.dm=0 ; di^ dt» dt* ^ „ d'^x , d^x „ , d'x . dt* dt^ di'- S«r'.rfm— 0 S.j/*.dm ^ 0 because x', j/, &c. are the coordinates of the body referred to the centre of gravity, see No. 15, page 91. PART I— BOOK I. 16 1 consequently we shall have m.- — =S.Pdm : dt* m..^=S.Qdm; V,. ^j) m -=S.Rdm : dt* these three equations determine the motion of the centre of gravity of the body ; they correspond to the equations of No. 20, which relate to the motion of the centre of gravity of a system of bodies. We have seen in No. 15, that for the equilibrium of a solid body the sum of the forces parallel to the axis of x, multiplied by their distances from the axis of s, minus the sum of the forces parallel to the axis oiy, multiplied by their distances from the axis of z, should be equal to nothing ; thus we shall have =5. [ (0?+/) Q~(j/+y')- P-l 'dm ; (!•) but we have S. (x.d^i/—i/.d^x).dm= m.(x.d''y—y.d*x); in like manner we have S. (Qx — Py). dm=x.JQdm—yJPdm finally we have S. {x'.d'-y-i^xd^y'—y.d''x—yd''x'\ dm—d*y. S.x'dm—d^x. Sy'dm Y 162 CELESTIAL MECHANICS, by the nature of the centre of gravity, each of the terms of the second member of this equation vanishes ; therefore the equation (1) will become in consequence of the equations A, C dt^ J * By performing the multiplication, . (Qj>— P3»).cfm+S.(Q.«'— P/).cf»", .-. by substituting for the expressions S. P. «?»», S. Q.rfw, to which they are respectively eqtjal ds appears from the equations {A), and freeing the quantities d »«/, «?^«, «, y, from the sign S, the preceding equation will be changed into the following ». &.QL. dm—y. SP. dm + ^^. Sx'Jm + «. ■S-^-<^'«— ^- -V-'^"' -y.S.^.dm +S i^L^l-fj^X . rfm:^. S. Q.d*-3^. S.P.rf»« + S {Q.x*—Py*). dm, and omitting quantities which destroy each other, and also those which by the nature of the centre of gravity, vanish, we will obtain the equation this equation involves the principle of the conservation of areas, for if the forces which s<*cit the Tndlecuks arise from their mutual action, and from the action of forces directed towards fixed points, S[Qx' — Py.) dm=0. PART L—BOOK I. 163 this equation integrated with respect to the time, gives S. S ""'^y'—if-^^' \.dm= S.fiQx'—Py'). dt. dm ; the sign of integration /being relative to the time t. From what precedes it is easy to infer that if we make SJ{Q.x'—Pij'). dt. dm=N-y S.J\R!^—Pz'). ^t. dm= N'i S.f{R^—Q.z'). dt. am=N"i we shall obtain the three following equations I dt S S.l ^.dm=N;y. ^s) these three equations contain the principle of the conservation of areas j they are sufficient to determine * the motion of rotation of a body about its centre of gravity ; combined with the equations (A), they completely determine the motions of translation and rotation of a body. Y 2 * In our investigations relative to the invariable plane in the 5th chapter, we have seen that when a body or system of bodies are not solicited by any extraneous forces, the motion may be distinguished into two others, of which one is progressive and the same for all points, the other is rotatory about a point in the body or system, the first determined by the equation {A), and the second by the equation(£) ; by thus distinguisliing the motion into two others, we can represent with more clearness the motion of a solid body in space,for these two motions are entirely independent of each other, as is evident from the inspection of the equations which indicate them, so that the equations (A) may vanish, while the equations (B) have a finite 164 CELESTIAL MECHANICS, If the body Is constrained to turn about a fixed point ; it follows from No. 15, that the equations {B) are sufficient for this purpose ; but then it is necessary to fix the origin of the coordinates x', tf, z', at this point.* value or vice versa. The centre of the rotatory motion may be any point whatever, how- ever when we would wish to determine these two kind of motions it is advantageous to assume for this point, the centre of gravity of the body, because in most cases its motion may be determined directly, and independently of that of the other points of the body. Dividing the equations (^) by m, we may perceive by a comparison of the resulting expressions, with the equations of the motion of a material point, which have been given in No. 7, page 31, that the motion of the centre of gravity is the same, as if the entire mass of the body was concentrated in it, and the forces of all the points and in their respective directions were applied to it ; this rectilineal motion is common to all the points of the body, and the same as the motion of translation. * If a solid body is acted on by forces which act instantaneously, in general it acquires the two kinds of motions, of translation and of rotation ; which are re- spectively determined by the equations (^A) and {B) ; when the equations (/^ ) vanish, the forces are reducible to two parallel forces, equal, and acting in opposite directions, when the rotatory motion vanishes the instantaneous forces have an unique resultant passing through the centre of gravity, see notes to page 143, when the molecules of the body are solicited by accelerating forces, their action in general will alter the two motions which have been produced by initial impulse, however if the resultant of the accelerating forces passes through the centre of gravity of the body, the rotatory motion will not be affected by the action of these forces, this is the case of a sphere acted on by forces which vary as the distance, or in the inverse square of the distance from the molecules, see Ne^vtou prin.Vol. 1 . Section 1 2, or Book 2, No. 12, of this work, consequently if the planets were spherical bodies, the motive force arising from the mutual action of the sun and planets would pass through the centre of gravity, and the rotatory motion would not be affected, but the direction of this force does not always pass accurately through this centre, in consequence of the oblateness of theplanets, therefore the axis of rotation does not remain accurately parallel to itself, however the velocity of rotation is not sensibly affected, see Systeme du Monde, Chapter 14, Book 4, and Books, No. 7 and 8. It is in this slight oscillation of the axis of the earth arising prin- cipally from the attractions of the sun and moon, that the phenomena of the precession of the equinoxes and of the nutation of the earths axis consist. (See Nos. 28, 29. If the body be moved in consequence of initial impulses, the directions of the forces, their intensities and points of application been given, we might by the formula of No. 21, de- termine the principal moment of the forces with respect to the centre of gravity, and the direction of the plane to whicli this moment is referred, which would completely determine the moment of rotation obout the centre of gi'avity, and it is evident that the same data would be sufficient to determine the rectilinear motion of the centre of gravity, and consequently the motion of translation of the system, see No. 29. PART I— BOOK I. 165 26, Let us attentively consider these equations, the origin of the coordinates being supposed fixed at any point, the same or different from the centre of gravity. Let us refer the position of each molecule to three axes perpendicular to each other, fixed in the body, but moveable in space. Let fl be the inclination of the plane formed by the two first axes to the plane of x, y, ; let (p be the angle formed by the line of inter- section of these two planes and by the first axis ; finally, let \|/ be the complement of the angle which the projection of the third axis on the plane of x, y, makes with the axis of x. We will term these three axes principal axes, and we will denote the three coordinates of the molecule dm, referred to those axes by x', y", z", ; then by No. 21, the following equations will obtain x'=x". (cos. S. sin. ^. sin. ip+cos. ^. cos. 9)4- y". (cos. 6. sin. vj/. cos. q> — cos. ^. sin. (p) + z". sin. 6. sin. ;J/ ; y = x". (cos. 9. cos. »}/. sin. (p — sin. vj/. cos. f) + y". (cos. 6. cos. 4'- cos. ?> + sin. \J/. sin. (p)-\-z". sin. 6. cos. ■^ ; a'= 2". cos. fi — y". sin. 9. cos. (p—x". sin. 0. sin. (p. By means of these equations, we are enabled to develop the the first members of the equations {B) in functions of 9, 4-, <P and their differentials. But this investigation will be considerably simplified, by observing that the position of the three principal axes depends on three arbitrary quantities, which we can always determine so as to satisfy these three equations. Say. dm= 0 ; S.x"z".dm=^0 ; S.y"z".dm=^ 0, * * In deducing the values of — N, — N', in functions of 6, t^, ip, and the coordinates x",2/", z", it is assumed that there are three axes possessing this property of having Sy'/z//.dm, Sx"y". dm=0, Sx"z"'. dm:=0. However it is afterwards demonstrated that there exists three such axes in every body. Since by hypothesis the principal axes preserve their initial positions, being moveable in space though fixed in the body, while the axes of x', y , and z', are fixed in spuce, it follows 166 CELESTIAL MECHANICS, thai let us make S. (/ --+3"*). dm= A ; S. (/'» + z"'). dm=Bi S. (^"*+y'*) dm Cj and in order to abiidge let us make tf(p — cfij/. COS. 6 = p.dt ; d^, sin. 6. sin. ^—d9. cos. ip =q.dt ; d^, sin. 6. COS. (p+d^. sin. (p= r.dt. The equations {J5) will, after all reductions, be changed into the three following ; A.q. sin. 0. sin. q> + Br. sin. 6. cos. (p'—Cp. cos. fl = — N ; Cos. i)/. [Aq. cos. 0. sin. <p-\-Br. cos. 0. cos. (p + Cp, sin. 0] + sin. 4^. [ Br. sin. ?> — Aq. cos. <p] = — N' ; ^i (Q Cos. ij/. [£r. sin. (p--Aq. cos. ?>} — sin. ^.{Aq. COS. fi. sin.ip+jBr. cos.6. cos. (p+Cp. sin.S] = — iV^" , that the coordinates «", y , z", are constantly the same for the same molecule, and vary only in passing from one molecule ^o another, but the coordinates s! i/ ^ vary witli the time .•. they are fiinctions of the time, as are also the angles 6, ■]/, <p, since they depend on the position of the principal axes with respect to the fixed axes .-. when we take the differentia! of jt', t/', and j/, with respect to the time in terms of x" y" z" and the angles d, ifj 'Pi we should not take the differentials of x",y' , s", it may likewise be observed that we can omit the consideration of those quantities of which one of the factors is the product of tvvo different coordinates, for such quantities disappear from the expression s/dy — y'dx', as they occur in the two parts of it affected with contrary signs, these considerations enable us x'dj/ i/dif to abridge considerably the investigation of the value of — , intermsof«"y'2" and fiinctions of the angles 6, 4", ^, for we shall not take into account, those terms which would eventually disappear in tfae expression x'dy' — y'.daf 7t • dt!= — x".[—d). sin. i.sm. i^sin. ?-f<f4- cos. 4» cos. i. sin. ^ -f (/<?. cos. (p. sin. 4- cos. ( — cfi^.sin. ■^. cos. ? — d(p. sin. ^. cos. i^) PART L— BOOK I. M7 tltese three equatdons give by differentiating them and then 'supposing 4/ = 0, after the differentiations, which is equivalent to assuming the j^y" ( — rfj. sin. i. sin. tJ/. cos. ?+ d-<|'> cos. 4'. cos. ip. cos. « — rf<p, sin. ?. sin. 4'. cos. H"<^'4' sin. ■v}'. sin. <p — rfip. cos. ip. cos. ij^) +2". (c?«. cos. «. sin. 4 + rfj/. cos. -i^ sin. «); dj/zzx''.{ — dK sin. tf. cos. ^. sin. ip — rfil/. sin. 4'. sin. <p. cos. i •{■dip. cos. ^.cos. 4. COS. 6 — di}/. cos. 4- cos. <p-f-c/ip. sin. <p. sin. 4) + y ( — dS. sin. *. cos. 4. cos. (p — c?^- sin. 4'' cos. ?>. cos. S —d^. sin. (p. cos. •4'. cos. 6-{-d-^. cos. 4'- sin. tp 4- rf(p. cos. 9. sin. 4) -^'.{dt.cos. 6. COS. ■4'— d-^' sin. 4- sin.«) «?/=: — !i'.dl.6\n.6—fy". di. COS. 6.jeos. :q>+y". dif>. sin. p. sin. 4 — y. cf«, cos. 6. sin. ip — a". dip..cas.Jl>. sin. « /. ^4/= (of', cos. S, sin. 4'. sin. ^ 4" ^^ oos..4'- cos. 9+;y '• cos, *. sin. 4'. cos. ip — .y. cos. 4*. sin. ip+z". sin. 6. sin. 4) X <— *"'rf<. sin. e, COS.4. sin, ^— «". rfif" s™- 'J'- sin. <p. cos. 6 + af' d<p. cos.^. cos. 4- cos. 6 — or" ^4- cos. 4- COS. ip+x" rfip. sin. (p. sin. 4 — -y . d6. sin. S. COS. 4- cos. (p — ^". d4. sin. 4- cos. 9. cos. i — ;y". «f^. sin. Ip. cos. 4- COS. ^+y. ^4- cos. 4. sin. <p-{-y". d(p. cos. (p, sin. 4 -f-z", cf^. COS. «. cos. 4 — «"• ^4- sin. 4- sin. 6) :r: — /'.* 6?^. sin. tf. COS. 6. sin, 4. cos. 4. sin.* (p — jf'.'^di. sin. }. cos. ^ 4- sin. ^. cos.(p — «".*rf4' sin. '4' sin. *<p. cos. ^6. — x".* d4. sin. 4 cos. 4- sin. (p. cos, 9, cos. « -}-x".' d(p. sin. ^. cos. <p. sin, 4. cos. 4- cos. 'S-\-x".^dip. cos. ^(p. cos. *4' cos. 6 -^x".^ rf4. sin.4'. cos, f^.isin. 9. cos. ^. cos.-< — x''.^d4'. cos. *4'' cos. ^<p +x",^df.sm.^ (p.sin. '^. cos. «+x".'rf^. sin. jp. cos. (p. an. 4* cos. 4. — n/'^.dt. sin. i. cos. «. sin, 4- cos, 4'- cos. *?i+y.* rfs. sin. «. cos. *4' sin. ^.cos. ^ —y".^d-^. sin, *4'« cos. '^ cos. ^t.+i/'.^d-^. sin, 4. cos, 4- sin. <p, cos. ^. cos. t 168 CELESTIAL MECHANICS, axis of x' indefinitely near the line of intersection of the plane of x'andt/', with that (£x' andy, — y'.* d^. sin. (p. COS. ^. sin. -i^. cos. -i/. cos. *^. +i/".^d<p. sin. ^(p cos. *\{'. cos. * +y.*rf4'.sin. 4'- cos.'vl'. sin. (p. cos. f. cos. * — y". ^d-^. cos. *'4'. sin. '^ + y.* dp. cos. ''^i. sin. ^■^. cos. « — y" .' d(p. sin. (p. cos. ip. sin. -^t. cos. ij/ ' -f'5'"'*.'^^' sin. *. cos. ^. sin. ■\'. cos. \J/ — z".^c?4/. sin.'^vj/. sin.*fl y.cfr'. = (i". cos. <. cos. ■J-, sin. ^ — x" sin. y'. 90s. <p-f-y . cos. *. cos. ■^. cos. <p -f-y". sin. -vj/. sin. (p-f-z'. sin. 0. cos. v|^) x ( — s". rf<. sin. J. sin. -i^. sin. <p + x". rfv}'' cos. i|'. sin. (p. cos. «-)-«". c?ip. cos. ^. sin. -i^. cos. « — b". cf4'. sin.'v^. cos. <p — al' .d<p. sin. <p. cos. -i^ — y", di. sin. tf. sin. ij/. cos. <p+y. d-i^- cos. -v^. cos. <p. cos. « — ;y", rfip. sin. <p. sin. -J" cos. « y". cfij/. sin. i^. sin. ip — y . ^ip. cos. <p. cos. ■v{/. + 2^'. di. cos. <. sin. i|/.-f-.~" d-i^. cos. ■4'. sin. (1)=: — i".*rf(i. sin. ^. cos. i. sin.il^.cos. ■vj^. sin. -(p — s".^d6. sin. «. sin. ^4^. sin. ip. cos. <p +«".» rfij'. cos. ^^^^ sin. 'Ip. cos. *«— «". V^-. sin. ■4'. cos. 4'- sin. ?. cos. tf>. cos. * "j-jt". *d^. sin, (p. cos. Ip. sin. \J/. COS. ■4'. COS. '6,— «".-rf<p. cos, '<p. sin. '\J/. cos. < '<f4'' sin. tJ'. COS. i^. sin. ^. cos. ip. cos. « ■(-x".'c?4'. sin. '^. cos. "^ 'rf^. sin. »<p. COS. ^•v}'. cos. ^-f-x". rfip. sin. (p. cos. ip. sin. \}/. cos. i^. —y''.'-d6. sin. ^ cos. «. sin. •vJ/. cos, 4'. cos. -<p — j/'.^di, sin. «. sin. 'tI/. sin. <p. cos, p +y,* d^f. cos, '4'' cos. =^. cos. *?i.4-!/''.'e^T^. sin ■4'- cos. i^. sin. (p. cos. $. cos. i. — ^y".*c(<p.sin. ip. cos. ?>. sin, ij/, cos. 4-. cos. '(. — y'.'^d'p. sin. -0 sin. 'tl'-cos. « 4-y .*rf<|'. sin. if" cos. 4'' sin. <p. cos. ip. cos. i-\-y" .' d-^i • sin, *4/. sin. ^ip — ^".*rfip. cos. *ip.cos. *4/. cos. ^— y.'rfip. sin. ?. cos ip. sin. •\'- cos. if) 4-i".^<f«. sin. 1 COS. «, sin. if/, cos. ■\'- +^"^(^^|'. cos. '4^. sin. *«. PART I.— BOOK I. 169 di. COS. 6. {Br, cos. 9 + Aq. sin. 9) +sm. 6. d. {Br. cos. (i>+Aq, sin. ip) — d. {Cp. cos. fi) = — cfiV^ ; <?>)/. (£r. sin. 9 — Aq. cos. ?>) — (fO. sin. 6. {Br. cos. ip H- ^y. sin. (p)+cos 9. d. (Br cos. <p + Aq. sin. ip) + cf. (Cp. sin. 6) = — (fA''' ; d. (Br sin. ip — Aq. cos. ?>) — d^. cos. 9. (£r cos. (p+ Aq. sin. 9) — Qj.rf" 4/. sin. 0 = -- dN" making Cp=p'', Aq= j'j 5r=/; z .". observing the terms which coalesce and those whicli destroy each other in the expression for y^iy — y^*') tWs function_becomes equal to — x".* dS. sin. I. sin. 9. cos.^— «".'rf4'' *'""• '?• cos. ^i — x''.^d<J/. cos. '^ +(«".» rf^. cos. ^ip. COS. 6 + a/'.* rf<fi. sin. '(p. cos. «) = {xf'.*d(p. cos. ».) +y.'t/J.sin. tf. sin. <p. cos. (f — y".*d-^. cos. '<?. cos. '< — if' .'^d'^. sin. *9 +(^".*c^ip. sin. ^(p. cos e-\-y".'d<p. cos. ^<p. cos. «) z: {i/'.*d<p. cos. «). — 3".V-v^. sin. ''«. This equation when extended to all the molecules of the body is identical with the equation, A.q. sin. $. sin. (p-\-Br. sm. 6, cos. 9 — C.p. cos. *. :£ — iV; taken with a contrary sign, for substituting in place of A, B, C, p, r, q, their values, in this equation, it becomes for one molecule , „ ,,, ( d-l>. sin. 't. sin. ^(p) — di. sin. i. sin. (p. cos. (p ") , (y + ^'Y I -^r— —5 + {x"^ + s"^). (d^f'. sin. »S. cos. "ip + tfO. sin, <. sin. ?>.cos. ^) — (^"* +«/''*) (<^ip. cos. 5 — dif'. cos. *«), _ equal by making all the quantities by which y,*z",' i",* are respectively multiplied coalesce so that they may be respectively factors of these coordinates 170 CELESTIAL MECHANICS, " these three diflPerential equations give the following ones * dp'+ \—-j—\q'r'.df=dN, COS. ^-—dN'.&m. 0; dr C—B CB A—C7 „, , dq'+ S i .r'p'.dtzz — (dN. sin. 9 + dN'. cos. 0). sin. ^ -\-dN". COS. (p ; h(i>) '+ ^ — j-rii 'P'q'-dt= — (dN. sin. 6 + dN'.cosJ). cos.^. — rfiV^". sin. (p. y.*rf^ (sin. ^«. sin. «(fi-|-tos. *^) — .y. -dd. sin. «. sin. ?. cos. (p — 7/". 'dip. cos. ♦ = i/''^.d4'. cos. '<f>. COS. -«-}-^",»c?t|'. sin.'ip — ^"*. rf«. sin. *. sin. (p. cos. (p~-i/'*.dp. cos, # r''^.^^". sin.*^. sin.*(p — 3"*.c?«. sin. «. sin. ip. cos. <p+3"*.c?4'. sin.»5. cos.*? -\-^'.'di. sin. 5. sin. <p. cos. (p = s*'. ^d^p. sin. 'S +«"* . d4'' sin.^^. cos. '^-{•x"^.d6. sin, «. sin. ip. cos. * — 3/'^.dp. cos. «-j-x'".(i'-i|'. cos.'S = a"'.rfT}/. (sin. ^ip. COS. *«)4-y ^(/•4/. cos. «ifi — x"\d/p. cos. e+x"'.d6.sm. «. sin. ip. cos. $, Since the angle -^ vanishes after the differentiations, wherever sin. ■v^ occcurs as a factor tliis quantity must be rejected, and wherever cos. ^^ occurs it becomes equal to unity, keeping these circumstances in view it will immediately appear that the expressions for — dN—dN" — dN" should be such as are given in the text. * The first differential equation being multiplied by — cos. 6 becomes equal to — dS. cos. '■6 {Br. cos. P + Aq. sin. ^) — sin. 6. cos. 0. d. (fir. cos. <P-\-Aq. sin. ip) + COS. (. d. {Cp. cos. e) = dN- cos. t and multiplying the second equation by sin. $, we have d^l/. sin. e.{Br. sin. (p — Aq. cos. (p) — d(, sin. '«. (Br, cos. p-j-Aq. sin. 9)+sin. t. cos. i. d.{Br. cos. ip+Aq. sin. ifi)+sin. S d.{Cp. sin. ()— — dN'. sin. d .•. dN. cos.« — d.N'. sin.«= — d6.{Br. cos.?i+ Aq. sin. (p)-\-d->p. sin. 6, (Br. sin. (p — ^y.cos.^) + cos.»«. d. ( Cp) — d6. sin. 6. cos. 6. {Cp)+sin. *6. d(Cp)+d6 sin. 6. cos. (.(Cp); = PART I.— BOOK I. 171 these three equations are very convenient for determining the motion of rotation of a body, when it turns very nearly about one of the principal axes, which is the case of the celestial bodies. 27. The three principal axes to which we have referred the angles z 2 by substituting for r and q their values — — B.{di.d-i^.'im.^. cosJ (p+clS.* sin. <p. cos. <p) — A-{de.d-^.sm. ^.sin. '<p — dS.'' sin. (p. cos, ip) . ___ + B.(rf'|'.* sin. ^6. sin. ip. cos. (p-j-di^.d). sin. 6. sin. -<?) — /4.(cAJ/.* sin. *(i. sin. <p. cos. (p _ —d4^. dS. sin. 6, cos. *0) , , _ . + d.(C.p.)= {B — /}.l(r/ij/.»sin. = ^. sin. <p. cos. <p) — dS. ' sin.tp. cos.(p+d4'.d6.(sin. ^.sin. "tp — sin. 6. cos. '^) ) _ + d.(C.p.)={B—A). q.r.dt+dp' = -^ . q'.r'.dt+dp' in like manner, multipl3'ing the first of the differential equations by sin. 0. sin. (p, the second COS. 6, sin. (p. and the third by— cos. <p, and then adding them together we obtain — dN. sin. 6. sin. (p — dN'. cos. 6. sin. <p — dN". cos. (p= to 6?«. sin. ^. cos. 6. sin. ip. (Br. cos. (p + Aq, sin. (p) + sin. *«. sin. <p. ^. (£r. cos. (p+Aq. sin. ?) — sin. 6. sin. (p. (/. (C^. cos. $) -\-d4'- COS. tf. sin. (p[Br. sin. ip — Ag. cos. ip) — cf^, sin. 6, cos. «. sin. (p(Br. cos. ip + /f y. sin. !p) +COS. =«. sin. (p. d. [Br. cos. (p+.^J'. sin. ip)+cos. 6. sin. (p. d. {Cp. sin. «) — COB. <p. d. (Br. sin. ^ — /4y. cos. ?i) + d^. cos. «. cos. <p. (Br. cos.(p+ /4y. sin. <p) + C^p. (f'<|'. sin. 6. cos. (p = by concinnating sin. Ip. rf. (Z?r. cos. <p-\-Aq. sin, (pj^-c^-vj/. cos. «.5r — cos. (p. d. (Br. sin. (p — y^y. cos. (p) — sin. «. COS. J. sin. <p. d.(^Cp) + £?*. sin.'S. sin. $. (C/))-|- sin. fl. cos. S. sin, <p. t/. (Cp) + de. cos. ^ ». sin. (p. ( Qj.) + f Cp.) d^^. sin. tf. cos. ip ; = sin. ip. cos. <p. c?. (/?r) + sin. *^. d. (Aq)—Br. d<p. sin. '<p. + Aq. dip. sin. ip. cos. ip. + rf^}'.cos.«. J5r— sin.ip.cos.ip. d.(Br)+cos.'(p.d.(Jq) — Br.rf^i.cos.* ip — Aq.d(p. sin.(p.cos.f. 4- rf^.sin. ip.(C.p.)+(C/;). rfvf-. sin. «. cos. ?i=rf.(^9)— B;■.rflp4-rf•^;'.cos.«.JBr -l-d^.sin.lp.((^),+<Z■vJ/. Cp.sin. ^.cos.ip) 172 CELESTIAL MECHANICS, S, p, 4'> deserve particular consideration ; we now proceed to determine their position in any solid whatever. From the values oi of 1/ z', which have been given in the preceding number we may obtain the following expressions by No. 21. x"-:z.af (cos. 6, sin. »J/. sin. <p + cos. ^. cos. <p) +3/'. (cos. 6. cos. »J/. sin. p —sin. ^. cos. 9) — z'. sin. 6. sin. cp ; y" = x' (cos. 6. sin. if/, cos. (p — cos. ■^. sin, 9) + ?/'. (cos. 0. cos. <|/. cos. <? + sin. i)/. sin. (f) — ~'. sin. 9. cos. (p ; ^"=j'. sin. 0. sin. ^■\-y'. sin. 6. cos. ^^-z'. cos. 6 ; From which may be obtained, x". cos. (p — y". sin. (p-=x'. cos. <|/ — y' sin. v)/ ; x". sin. (p+y. cos. (? = y. cos. 6. sin. 4'+j/'- cos. 0. cos. 4- — ^'- sin* S ; and making S.x'Mm=ia-; S.y'J" dm=b-; S.z'.^dm=c-; S, x'y.'dm—J'; S. x'z', dm—g ; S. y'z. dm —h ; we shall have COS. (p. S. x"z". dm — sin. (p. S.y"z''. dm= (a^'—b^) sin. 6. sin. vf/. cos. ^ but by substitution d.{Ag) + Br.{ — d<p-\- d-^. cos.e)+c?«. sin.ip. C p.-\-d^{Cf.)sm.i. cos. <p.z= , . . — Bd-^. dip. sin. 6.cos.ip — dip. dS.sin.(p-{-d^.~sm.6. cos. 6. cos.p+d4:di.cos,6.sin.<p. d.(Sq) -\-C.(d6. dp. sin. ip — df. d4^. cos. 6. sin.(p)4- C. d-^. dp. sin. 6.'cos, p — C.d^.- sin. S.cos.C.cos.^. _ ~{C — B).d<p.d4'. {sin. 6. COS. p.)-i- {dp. dS.sin. p) — rf-vj/. ^ sin. «. cos. 6. cos. 9 — d6. rfi|/. cos.Ssin. <p) dt +d[Aq.) ~ (C-B). p. r.dt + d.{Aq.) - ^^^^p'.r'dt + dg' CB by a similar process we might deduce the value of the last difiFerential equation. PART I.— BOOK I. 173 -^f. sin. e. (cos.2 v}/ — sin.^v|/) -|-cos. 9. {g. 008.4- — h. sin. ^) ; sin. <p. S. x'^ z" dm + COS. (p. S. 7/"z".dm= sin. 6. COS. 9. (a.^ sm.^+b.^cos.'^—(r + 2f. sin. >}/. cos. ^)* + (cos.^9— sin.'9;. (g. sin. ;}/+/«. cos. \j/). * r''. COS. $i=x'. (cos. *. sin. il'- sin. <p. cos. <p-}-cos. ■vj/. cos. *ip) +y . (cos. *. COS. 1^. sin. (p. COS. <p — sin. 4- cos. ^<p) — z'. sin. ^. sin. <p. cos. (p. y. sin. (pzzx'.lcos. 6. sin. •v}'- sin. (p. cos. ip — cos. 4'- sin. 'ip) +,y'. (COS. 6.cos.-<p. sin. *. cos. ip+sin.4'. sin. ^(p) — s'. sin. 0. sin. ip.cos. ip, .*. x". COS. <p — y". sin. <?=.!/. cos. ^}' — •?/. sin. ■4' x". sin. <p=x'.{cos. 6. sin. ■vj/. sin. 'iji-j-cos. •v}/. sin. (p. cos. $) +y (cos. ^. cos. 4*. sin. ^ip — sin. ■^. sin. ip. cos. ?i) — ~ . sin. ^. sin. -ip. y. cos, <p=x' (cos. 6. sin. 4'. cos. *ip — cos. i^- sin. ip. cos. ip) -j-y. (cos. (. cos.-il'. cos. *(p + sin. ■^. sin. (p. cos. <p) — z'. sin. «. cos. 'ip .*. .t". sin. (P-)-^". cos. (p=x'. COS. ^. sin. ^-{-y'. cos. «. cos. 4 — /. sin. tf ; multiplying the first member of the equation x". cos. (p — y" . sin. ip=x'. cos. 4 — y* sin- '4" by z" and the second member by the value of z" we obtain cos. (p. x"z" — sin, ip.y .z"=: a'.^ sin. i. sin. •4- cos. -^—afj/. sin. «. sin. '4' -f-a^y. sin, C. COS. ^4 — y-* sin. ^. sin. 4- cos. 4'- +2;'. i'. COS. 6. COS. 4^ — s/^'- COS, 6. sin. 4'j substituting for «',* y',* j;'y, z'y, 2'x', their values and concinnating we obtain COS. ^.a:"!" — sin. ip.t/' z!'z={.x'- — ^"). sin. S. sin.4'- cos.4' + ^y- sin. 6. (cos. ^4^ — sin.'4'') + 2'j'. COS, 6. COS. 4 — ^y- cos. 6. sin. 4> this expression being extended to all the molecules of the body, will give by substituting for S-r/^dm Sj/.^dm,&c. their respective values a-,b^,/,g,h, &c,the expressionm the text, in like manner sin. (p. x" z" 174 CELESTIAL MECHANICS, by equalling the second members of these two equations to nothing, we shall obtain . /?. sin. x|/ — ";. COS. J/ (a^ — b'^y sin. \j/. cos. ^■\-J. (cos.'^ij/ — sin."* ■i/) J ^. sin. il/+/;. COS. »|/ 2" tan. 2? „ ., • o , , o u , ,^ ,. : — i x~ ' "^ r — a-. Sin. "4/ — 6 . cos.-.\}/ — 'ilj. sm. »J/. cos. 4' but we have always tan. 6 i tan. 26 = 1— tan.-°fl ' by equalling these two values of tan. 29, and substituting in the last ex- pression, in place of tan. S. its value, which has been given in a function of <j/ J and then in order to abridge, making tan. i}/= z< ; we sliall obtain after all reductions, the following equation of the third order.* 0=(gu+h). (Jm—gy + [ {a-—h^). u-vf. (1— zr) ]. {h(^—Jia'+fg). u+gb'—gc'-hj] . 4-C0B.?i.y.z".=:a;'' sin. 6. cos. 6. sin.*Aj/+,fy. sin. «. cos. «. sin.'>}'.cos.4'— s'«'-sm. = «. sm.>J' -f-a't/'. sin. 6. cos. 6. sin. 4'- cos. ^+1/'." sin. 6. cos. 6. cos. '4'~^!/ *'"■ "*• cos. i^ + ;V. cos. ^6. sin. -^-^z'y'. cos. *«. cos. ij/— z'.' sin. «. cos. 6 ziz sin. «. cos. 6. (a'.'* sin. °^+y.» cos. '-^ — z'^)+'2x'y'. sin. 6. cos. «. sin. ■4'. cos. 4') + (cos. ««— sin. ««.) («'*'. sin. T^.-f^Y- cos.^}..) by extending this expression to all the molecules and substituting a^,i*, c', /',y5, &cact. tor Sx'V»K and S/ ^ f/w &c. we shall obtain the expression which has been given in the text. * The second members are put equal to nothing because by the conditions of the problem, the first members respectively vanish, consequently we have 0 = ( (flS— 6*). sin. ■4. cos. 4+/. (cos^^— sin. ^4)). sin. 6-\-(g. cos. 4— /(. sin. 4). cos. < ; 0*- sin. 6. cos. i. (a- sin. ^-\-b? cos. ^ — c2+2/ sin, ■^, cos. 4) + (cos. !'«— sin. H). (g. sin. 4+/'- cos. -f) ; PART I.— BOOK I. 175 As this equation has at least one real root we may perceive that it is always possible to make these two subsequent expressions, and con- sequently the sum of their squares, to vanish at the same time sin. 6 h sin. 4' — g- cos. ^ •■• ^^e" *^"" * ~ (««— 6-). sin. 4..cos.4'+/(cos.2^^— sin.«^J.)' sin. e sin. <■ cos. 9 _ COS.0 ^!^=: i tan. 2(» = COS. *(»— sin. «* — sin. ^6 l—UmM COS. -i g. sin. ■^.-\-h. COS. -^ (? — a.-sin.24/ — 6.2cos.-^J' — 2/ sin. ^. cos. i^ lhe«e fractions being divided cos. ij', become by substituting u in place of sin. •vj/ hu — g gu-{-h COS. 4'' ((a2_6»). u-{-f. (1— u-)). cos. 4- ' ( (c- (l+;r)— «- tfi—t^—'lfu).cos.-^ if we call the factors of cos.ij/ in the denominators of these respective fractions m and n we shall have tan. «.= "~^ . .-. i. tan. 2« = ?w. cos. y 1 C ^"-g 1 ' = m. cos.>V -hu-y ~ «• c°«- ^ •■• C_hu-g_l (_m. cos.^^J by reducing we obtain {hu—g). »m. cos. 'i^ = (gtt+h). ( .(m cos\|'f — (Au— f )') and consequently 0 = cos. ^. (m. (hu—g). n— (gM+A).>»)+(A«— ^).* (^«+^,) now {hu—g)n.=>. by substituting for n, {c?Q.-\-u^) — a^u^—h^ — 2fu) and then multiplying h(~ u+h(? 1^—ha^ uS—hH u—2fhi^—g(?—gc' ui+gahi^+glr'+^/gu, in like manner 176 CELESTIAL MECHANICS, COS. (p. S. x"^'.dm — sin. ^. S. ^'.z".dm ; sin. f. (S.yy.fi^wj + cos. (p. S.y"z'.dm'^ and this requires that we should have S, x"z».d'm ; Sy"z".dm separately equal to nothing. The value of u gives that of the angle 4'j and consequently the value of tang. 6, and of the angle 6. It is only now required to determine the angle <p and this will be effected by means of the condition S. x''y".dm =0, which we have yet to satisfy. For this purpose it may be observed, that if we substitute in S.af'x/'.dm * in place of x", y", _(g«+A). w. = {-(gu + h).{ai-b^). «-f-/(l-M=) = —a-gu^+gb^u^—gfu+gfus—ha^u+h b^u—hf-^hfu^ .-. the preceding equation becomes, to by making the similar factors of ic and its powers to coalesce, equal to, ^_ ' {h<? u. (1 W)-1'ai u. (1 + ui)-fk. ( 1 +2i.')-gc? (l+M2)+/g«, (!+«') iorcos.H.m.ihu-g) j£^£2:lWlll) + (hu—g)2. igU + h) = ( (a'-b.f M+/(l-«=)). {(hc^-ha''+fg).u-/h-gc^+gb'') )+(hu-g).^{gu+h)^ 0, which is the expression given in the text. * x". COS. (p — y". sin. ip=x'. cos. ■<J' — y. sin.'xj'. = P i". sin. ^-j-y. cos. (p=x'. cos. i. sin. ^-\-y'- cos. 6. cos. -^^ — z. sin. 6 =Q .'. «". cos. ■*?— y. sin. <p. cos. <p=s'. cos. ■4'. cos. (p— y. sin. i^. cos. (p=P. cos. (p x". sin. ^if>+i/'. sin. ?>. cos. ip = x'. cos. J. sin. ^. sin. ?-j-y. cos. «. cos. ^. sin. f — z' sin. «. sin. ^ = Q. sin. ^ .-, jr". = x'.(cos. t. sin. ■\J'. sin. <p + cos. if., cos. ip) +y (cos. «. cos. i|.. sin. (p— sin. ^|', coe. p) —J. Bin. *. sin. ^= P. cos. ^+ Q. sin. ^ PARTI.^BOOKI. 177 their preceding values, this function will assume this form, H.sin 2(p+L. COS. 2(p ; H and L being functions of the angles 9 and ^, and of the constant quantities a^ b^, (?,f, §•, A, by putting this expression equal to nothing, we shall obtain tan. 2ip = — — . The three axes determined by means of the preceding values of 9, 4^ j and <p, satisfy the three equations, A A xM. sin.ip. cos.$ — y". sin. *(p=x'. cos. 4'. sin. ip — ;y'. sin. 4'sin. <p=zP. sin.(p x". sin. <p. COS. <P-\-^' COS. ^(p=x'. cos. 6. sin. •J^. cos. ^-{•y'- cos. tf. cos. ■^^ cos. ^-:- :' sin. 0. cos. 9= Q. cos, 9 .'. y =s'.(cos. «. sin. ^. COS. $ — cos. 4^. sin. (p)-fy(cos. «. cos. if'- cos. (f+sin.rj'. sin. <p) +^' sin. «. cos. (p=Q. COS. ^ — P. sin. <p ,♦. x"y = PQ. COS. *$ — PQ. sin. '<p + Q.- sin. <p. cos. ip — P.* sin. <p. cos. <p .'. if Sx/'i/'. dm = 0, we shall have SPQ. dm (cos. «.p— sm. ■-?)) + S(Q'— P»).rfm sin. <p, cos. <p = 0. and -^l^-H^ sin. (p. cos. (f cos. *(? — sin. 2^, ' making H = S(Q'— P«)«?m and 21,= S.PQ rfw we shall have— ^-^^ sin. 2^. --// J5 -77; •'. — zf~ tan.2ipj 2. cos. 2^ H this equation determines a real value for, tan. 2ip and .•• for (p, and as the equation which de- termines the value of u has at least one real root, tan. 4' and .•. tan. 6, are real, consequently we are justified in assuming as we have done S,x//y'fdm, Sfil'dm, Ssf'ti'jdm, respectively equal to nothing, and therefore we shall have at least one systenj of principal axes existing in every body. 178 CELESTIAL MECHANICS, S.x'y.dm—0'y syz".dm=iO'y &yV.</m=0.* The equation of the third order in it, seems to indicate three systems of principal axes, similar to the preceding ; but it ought to be observed * All the roots in the equation which determines the value of u are real, and this equation must be of the third dimension, for in the investigation of the angles (, il', <p, there is no difference between tlie principal axes, nor is there any condition to determine which of the three principal planes we assume, ••. the solution must be applicable equally to the angle contained between the axes of a/, and either of the three intersections formed by the plane of jr', y , with the three principal planes of the body respectively, consequently the roots of the equation must be all real, it also follows that there is only one system of principal axes in every body, for as each system would give three values of u, the dimension of the resulting equation which determines the value of u, should be equal to three multiplied into the number of systems, but the equation does not transcend the third order, .•.the number of systems is only one, indeed if the equations which give the values oi i -^ and ^ are identical, tlie number of principal axes is infinite, tliis will evidently be the case where the tierms which compose the equation in u vanish without supposing any relations existing between the terms i, e, when a * — i ^ =c * , andyj g, h, respectively vanish we shall have for the coordinates a/,T/,ii, S.x'i/. dm =;0, S./cVf/m— 0, S.y ^ .dm=.0 .'. they are principal axes, and as in this case tan. (zz — ,the position of these axes is entirely undetermined .-. all systems of rectangular axes are prin- cipal axes and their number is infinite ; from the expression for tan. 6 it appears in like man- ner, that this angle is 100°, when a^z^b* and_/— 0, and consequently that the plane of the axes of ;/' and x' must pass through the axis of z". For all bodies symmetrically constituted, one of the principal axes, is the axis of the figure i, e, a line perpendicular to the plane dividing the bodies into two parts perfectly equal and similar, for supposing this plane to be that of x, y, then if we take two equal molecules, similarly situated ^vith respect to tliis plane, it is evident that if the coordinates of one molecule be X, y, z, the coordinates of the other will be jr, y, — z, and the indefinitely small elements of the integrals S,xz.dm, S._y2.f/n!, which correspond to these molecules will be a;z.rf»?, — xz dm,yz.dm — yzAm, .•. the sum of all the indefinitely small quantities xz.dm, — xz.dm, yzJbn — -yzydm, at one side of the plane will be equal to the sum of the indefinitely small quantities at the other side affected with a contrary sign, . ■ . their resjiective aggregates S.xzdm, S.yz.dm are equal to nothing, .'. the axis of z is a principal axis, and if the molecules of the body be sjTnmetricaUy arranged with respect to a plane passing through the axis of z' perpendicular to the first mentioned plane, we shall have S.xy.dm =:0 .-. the axes of x,y, z, will be prin- cipal axes. What has been established in the preceding note is of great importance, as the investi- PART L— BOOK I. 179 that u is the tangent of the angle formed by the axis of .r', and by the intersection of the plane of :v' and 3/' with the plane of x''' and y", and it is evident that one of the three axes of of', of y", and of z" may be changed in another, since the three preceding equations will be always satisfied j therefore the equation in u ought to determine indifferently, the tangent of the angle formed by the axis of of, with the inter- section of the plane ^', 3/', either with the plane x" y \ or the plane ocf\ z'\ or finally with the plane y", z",. Thus the three roots of the equatiffii in ic are real, and they belong to the same system of axes. It follows from what precedes, that generally a solid has only one system of axes, which possess the property in question. These axes AA 2 gation of tlie position of the principal axes is considerably facilitated by making one of them to coincide with one of three coordinates x' t/ z', whose position is entirely arbitrary, for sup- posing the axis of x" to coincide with the axis of xf, then since (p:=the angle which the intersection of the plane of x" and y", with the plane 3^,1/, makes wiih the axis of «', and since -^ = tlie complement of the angle, wliich the projection of the third axis on the plane of x' and y' makes with the axes of *', these angles are severally equal to notliing tan. i ^. sin. -J/ — g. cos. 1^ (a' — 6 ») sin. ^. cos.'vj/-j-y;(cos. ^-^ — sin.'^) becomes equal to _llandi tan 2«-— S:iiili±i:i^!^:i - _^lj_ /' J- • — c»_a2. cos.^4— 6^sin.'i^— g/sin. 1^. cos. i^ ~ c'*— i'« ' n which (/, 6',/', g', A' indicate what c, i,7>^j Aj become when a;' coincides with x", and as tan. 2[S + 100)— tan. (2(1+200) = tan. 2«, it follows that the other two axes must be taken in the plane y, 2', one making the angle i and the other the angle fl -f 100 with the axis of y, now if we made the axes of y", and s", to coincide with the axes of y and z' respectively, 6, and .•. /;' would vanish, and consequently S{y'z'.)dm would be equal to nothing. But if h' remaining equal to nothing, b' and c' would be equal to each other then // 0 tan.2« = — — p-j would be equal to — .•. « would be indeterminate and every line in the plane y' z', and passing through the origin of the coordinates would be a principal axis, see notes to page 184. 180 CELESTIAL MECHANICS, have been named principal axes of rotation, on account of a property which is peculiar to them and which will be noticed in the sequel. The sum of the products of each molecule of the body, into the square of its distance, from an axis, is called the momeiit of inertia of a body with respect to this axis. Thus the quantities A, B, C, are the moments of inertia of the solid, which we have considered, with respect to axis of -f", of 1/", and of ^'. Naming C the moment of inertia of the same solid with respect to the axis of z', by means of the values of y, y, and z*, which are given in the preceding number, we shall find C = A. sin.* 9. sin.« (p+B. sin.* 9. cos.* ip+C. cos.* 6. * The quantities sin.* 0. sin.* <p, sin,* 9, cos.* <p, cos.* 9, are the squares of the cosines of the angles, which the axes of x^^, of y, and of z'', make with the axis of z' ; hence it follows in general that, if we mul- tiply the moment of inertia relative to each principal axis of rotation, * Since &(j/'^+y24-j5"2).(^m=:S.(x'^+y^+2'^). dm by substituting the value of ^'r in terms of a.-",2y,2~".- and observing that S^'y"Jm, Sx"z".dm, Sj/'z".din, are equal to nothing, we have Si!'.''dm^S^'.'dm-!(.Sz".-dm=S^'?dm-\-S.y'.^dmJ{.S.x!'.^ sin.^^.sin. ^(p dm -\-S.y" .- s.m.-6.co%.-^.dm+Sz!' ? coih.dyn .: Sx"%l—sin.-6. sm.'<p)dm +Sy". (l_sin.2«. cos.-p).dm+ S.:^'.^(l—cos.^e).dm = S{t/^-\-t/^).dm .-. = S.a/' .^ (cos.^d+sm.\ cos.^<p).dvi +Sy.\cos.^6+sm.^6. sm.^(f>)dm +S.Z//.* sin.*tf. sin.2<p dm + S.«".*sin.2«. cos.-Um and making the like factors coalesce we obtain C S.{y".^+^y sm\sm.^<p.dm+S.{3/'^+z"^).sm.^6. cos.'<p.dvi+ S (x"^+f^).cos.H. dm i, e, C = A. sm.^6. sin.^i^+£. sin.^«. cos.*<p+C. cos.''«.; sin, 6. sin, tp, sin. 6, cos. (p, cos. 6, are equal to the cosines of the angles which the axes of x", of y and of z" make with the axes of z', see Note, page 132. PART I.— BOOK I. 181 by the square of the cosine of the angle which it makes with any axis, the sum of the three products, will be the moment of inertia of the solid, relative to this last axis. The quantity C" is less than the greatest, and greater than the least of the three quantities * A, B, C, ; therefore the greatest and least moments of inertia appertain to the principal axes, t * Let A be the greatest and C the least moment of inertia, the value of C ' may be made to assume the following form C'= A + {B—A). sm.'^d.cos.^(p-\-{C—A).cos.% . ■ . since the moments of inertia are always affirmative, the two last terms of the second member of this equation will be negative, consequently C is less than A, let C be the greatest moment of inertia and the expression for C will become C + (A—C). s.ia.-6. sin. ^(f>+(B— C). sin.^«. cos.2(p, in this case also the two last terms of the second member are negative, . • . C is less than C ; the moment of inertia C is greater than the least of the three principal moments, for if A be the least of the three moments which refer to the principal axes, we have as before C'—A + {B—A). sin.s«. cos ■><!> + (€— A). cos.2«, and as the differences are on the present hypothesis affirmative, C is greater than A, let C be the least of the three moments, and we have C'=C+(/l-^C). sin.^O.sin.'?! + (B—A).sin.\ cos.V, the terms which compose the second members are always affirmative, . ■ . we conclude that C is greater than the least of the three moments, A, B, C, From what has been established in the preceding note, it appears that when the three principal moments of inertia are unequal there is only one system of principal axes, for let there be another system and make A', D', C, the moments of inertia relative to these axes, then we shall have at the same time A "^ A' and A' "^ A which is impossible, see note to page 178. t For S{J—XY.dm^Sx'.^dm—2X.Sx'.dm-\.X^m—S3;.^dm—21!^-\-X^mM Sx'.dm = X.m. and as the quantity— w. (X^+ Y^) is essentially negative, the moment of inertia witli respect to the centre of gravity must be less than the corresponding moment for any axis not passing through the centre of gravity. If the moments are referred to an axis passing through a point different from the centre of gravity and of which the coordinates are a, b, c. 182 CELESTIAL MECHANICS, Let X, Y, Z, be the coordinates of the centre of gravity of the solid, relatively to the origin of the coordinates which we fix at the point about which the body is subjected to revolve, if it is not free ; x'-^X, y' — Y, z'—'Z, will be coordinates of the molecule of the body, with respect to the centre of gravity ; therefore the moment of inertia, relative to an axis passing through the centre of gravity, and parallel to the axis of zf will be s.^(x'-x)*+0'—Yy } dm: but from the nature of the centre of gravity, we have S. x'.dm=mX, S.y'.dm=mYi .'. the preceding expression will be reduced to Consequently we shall have the moments of inertia of the solid, with respect to an axis passing through any point whatever ; when these moments are known for axes passing through the centre of gravity. At the same time it appears that the minimum minimorum of the moments of inertia appertains to one of the three principal axes, passing through this centre. Let us suppose the nature of the body to be such, that the two moments of inertia A and B are equal, then we shall have C'=^. sin. *e+C.cos.'6: * the value of the moment of inertia with respect to this point is equal to It is evident from an inspection of their values, that the greatest moment of inertia with respect to any point, is less than the sum of the other two moments. * When A=B the moment of inertia with respect to any other axis = A . sin.^f + C. cos.^l, and as neither 4- or (p occur in this ejcpression, the moment of inertia for all axes making the same angle, with the axis of z are equal, and if « be a right angle C-:z.A, therefore in this case there is an indefinite number of principal axes, but they have all a common axis z\ when «=100* we have a^=-h^ and/= 0 i, e, Sx'MmzzSi/.-dm and Sxy.dm=0 this also PART I— BOOK I. 183 and by making S equal to a right angle, wluch will render the axis of ^ per- pendicular to the axis of z", we shall have C=A ; therefore the moments of inertia relative to all axes situated in the plane perpendicular to the axis of z'' are then equal to each other. But it is easy to be assured that we have in this case for the system of the axis of z^', and of any two axes perpendicular to each other, and to this axis, S. afy'.dm= 0 ; S. afz«,dm= 0 ; S.i/'2^'.dm= 0 ; for if we denote by x" and y the coordinates of a molecule </m referred to the principal axes, taken in the plane perpendicular to the axis of 2", and with respect to which the moments of inertia are supposed equal, we shall have or simply S,x'.*dmz=.S. y"*.dm; but by naming i the angle which the axis of z makes with the axis of af', we have x'=^ x'l. cos. £+y. sin. i ; y = yi'. cos. e — :>/'. sin. t. ; consequently we have S. x'y'.dm = S. x"y".dm (cos.»£ — sin.' i) 4. S. (j/"«— .or"^). dm. sin. i. cos. 1 =0 we shall find in like manner .S". .rV.rf?w = 0; S.T/sf.dm-^O'f therefore all axes perpendicular to the axis of z", are in this case principal axes ; and in this case the solid has an infinite number of similar axes. follows from the equations x':=x". cos. e + ^° sin. e, y =r y". cos. i — x ' sin. s for S. x'Mm ■=. S.{x" ? cosS-\- Sy". - sm..^i). =S.x"?dwz:Sy'. -dm, since Sx"y".dm-=zO, in. the case of an ^ipsoid generated by the revolution of an ellipse above its minor axis, we have always two of the principal moments of inertia equal, the moment which is the greatest is referred to the minor axis. 184. CELESTIAL MECHANICS, If we have at the same time A = B=C; we shall have generally Cz=.A ; * that is to say, all the moments of inertia of the solid are equal, but then we have generally, S.x'y'. dm=0; S.x^.dm=0 ; S.i/'z.dni= 0 ; whatever may be the position of the plane of x' and of i/' ; so that all the axes are principal axes. This is the case of the sphere, and we shall see in the sequel that this property belongs to an infinite number of other solids of which the equation will be given. t * Since by hypothesis ^=i?=C, Sxff.-din=Sy"Mm=SzV.2dm, .-. if in the expression for 2'2 in terms of :r//,°y',- zV and of the angle 6,4'><P, we take this into account and also observe that S.x'y". dm, Sx"z".dtn. S.if"z".dm, are equal to nothing, we shall find S.z^ dm= Sz'.'dm for z'=z". cos. 6— y". sm. ^. cos. <p—x". sin. 6. sin. :p .-. z'^ = z".- cos.^«+y'."sin.^«. cos.^(p-{-x'.- sin.^^. sin."(p=(when x"' =y'- =2"^) z'? the same is true re- $pecting y'^ and x^ on the other hand if we equate z"^ and its value in a function of x y, z and the angles ^, ^, ^, and also satisfy the equations Sx'.^ dm=Sy'.~ dm^Sz'dm, we must equate Sx'y'.dm, Sx'z'dm, Sy'z'dm to notliing. (See Book V. Chap. I. No. 2.) t x'' y" z" being the coordinates with respect to the principal axes of any point of the solid, if we transfer the origin to a point of which the coordinates are a, b, c, then the coordinates relative to the new origin will be «" — a,y" — b, z" — c, now if we suppose that the three principal moments of inertia with respect to this new origin are equal, then all rectangular axes, and .•. the axes of ««' — a,y" — b, z — c, will be principal axes, consequently we shall have 2.(a/'— a) (y"—b).dm - l..x"y".dm—a 2.y".d7n—b ^.x"dm + a b 2(/»! = 0 2.(x"—a).{J'—c).dm = ^x"z".din—a 2.z".f/»i— c ^x".dm 4- a c. 2c?jh =0 , 2.(y'— 6).(2"— c).rfTO = 2.y".z".dm—b 2.z".d»i—c 2.y".dm+bc 2dm— 0 now if we suppose the origin of the coordinates x",y", z", to be at the centre of gravity the preceding equations will be reduced toab. 2dm=0, a c.Zdm=0, be. l,dm=0 . •. two of the preceding quantities must vanish, let b, c, be equal to notliing and a will be unde- termined, .•. the point required wll be at a distance equal to a from the origin by a fore- going note the moments of inertia with respect to this point will be A,B-\-ma-, C'-j-?Ha'and by the conditions of the problem they are supposed to be equal .•. we have a= + \I'A—--(I I 1 •■• -.^ being greater than C we have two values of a equally distant on PART I.— BOOK I. 185 28. The quantities p, q, r, which we have introduced in the equations (C) of No. 26. ha e this remarkable property, that they determine the position of the real and instantaneous axis of rotation with respect to the principal axes. In fact, we have relatively to all points situated in the axis of rotation, ^/a.' = 0 ; c?y=:0 ; dz' = 0; if we difference the values of x', y', z', of No. 26, and then make sin. 4' = 0 after the dif- B B opposite sides from the centre of gravity, but a is also equal to V .•. in order that these two values of a should be possible, it is requisite that B should be equal to C, .', when A B C axe unequal there is no point vi^hich satisfies the required conditions and when two of the moments are equal, the tliird must be greater than either of them, and in this case the point required is situated on the axis relative to which the principal moment of inertia is the greatest, when the tliree moments of inertia are equal the two points are concentrated in the common centre of gravity. Wlien BzzC we have S.y".''dmzzS^'*dm. In an ellipsoid generated by the revolution of an ellipse of an ellipse round its minor axis two of the three principal moments relative to the principal diameters are equal, and the greatest moment is relative to the minor axis, see note page 181, .-. we shall have two points existing on this axis relatively to which all the moments of inertia are equal, it is easy to shew that the distance of those points from the centre of the ellipsoid is = to the square root of the fifth part of the difference between the squares of the semi-axes, and .-. they may be within the ellipsoid, at its surface, or finally without tliis surface. We might have inferred a priori that there is an axis with respect to which the moment of inertia is a maximum and a minimum, for from their nature all moments of inertia are positive and have a finite magnitude, and most authors deduce the properties of principal axes from the moments of inertia which are the greatest and least, the general expression for S.[i"^ -\-y"').dm in terms of x' i/ and z! is equal to SJ^. dm. COS. '■i sin. ■■^-\-S.x'^dm. cos. '■■^-{■S.i/^dm cos. ^L cos. •^■i-Su\^dm sin. '^ + S.z'^.dm sin. 'rf -(-2 Sx'i/dm. cos. ^6. sin. i^. cos. i^ — 2S.x'i/'.dm. sin. i^- cos.-4/ — ^S.z'x'.dm sin.6. cos,6, sin.\f/— 2 S-z'^/dm. sin. i. cos. i. cos. if. When the law of the variation of the density and the equation of the generatino- curve of a solid of revolution are given, the value of S.(x' *+?/''). dm may be computed by a method similar to that by which the centre of gravity of a body is determined ; the value of S(x'-j-y*).rfm is computed for the earth in Book V. Chapter 1. No. 2. 186 CELESTIAL MECHANICS, ferentiations which we are permitted to do, since the position of the axis of x' on the plane of x, y', is indeterminate, we shall have dx'=x",\d^. COS. 6. sin. <f — d(f. sin. (f)+y". {d^. cos. 6. cos, 9 •^d(p. COS. (p\ -\-z". d-^. sin. 6=0 ; di/'^x". \d(p. COS. 8. COS. 9 — d^. sin. 6. sin. (p — d^. cos. ^] + y. \d^. sin. <p — d(!). cos. G. sin. <p — rf9. sin. 0. cos. (p\ + z". </fl. cos. 6 = 0 ; d^ = — x".(^dL COS. 9. sin. (p + d<p. sin. 6. cos. ip) — y* C*^^' cos. 9. COS. (p—d(p. sin. 6. sin. (p) — 3".r/9. sin. 6=0. If we multiply the first of these equations by— sin. 9 ; the second by COS. 9. cos. p, and the third by — sin. 6. cos. <p ; we shall have by adding them together, Multiplying the first of the same equations by cos. p ; the second by cos. 6. sin. (p, and the third by — sin. 6. sin. <p, and then adding them together we shall obtain 0=pf — rz". Finally, if we multiply the second of those equations by sin. 6, and the third by cos. 6, and then add them together, their addition will give * . 0=qj/' — rx." * In taking the differentials of dxf, dt/, d^, we may omit those quantities in which sin. 4- occurs after the differentiations, and where cos. ■^ occurs, we may substitute unity ; multi- plying the value of dx' which results by — sin. <p, it becomes — ttc'. sin. 9 = — x".(rf4'- COS. 0. sin. ^<p — d<(i. sin. *(p) — y" .{(l-^.co&, t, sin. ^ cos.f —d<p. sin. 9. COB. 9}— z". d-^. sin. k sin. <; ; PART L— BOOK I. 187 This last equation evidently results from the two preceding ; thus the three equations dx'=0, dy'^=0, dz' = 0 reduce themselves to these two equations which belong to a right line, forming with the axes of x^\ B b2 and in like manner multiplying dt/ and its value by cos. 6. cos f, we liave di/. COS. ). COS. i?3:a"(rfip. cos. "6. cos, *^ — dS. sin. 4. cos. isaa. f. cos. (p — d^'. cos. *.cos. *?) -\-i/'.(d-^. sin. (p COS. ^. cos. i — d<p. cos. "-(. sin. ?. cos. (p—d6. sin. L cos. 6, cos. *f ) +2;".rf«. cos. '«. COS. ^ and the multiplication of dz!, and its value by — sin. 6. cos. (f, gives — «?z'. sin. 6, COS. ip = x"{dS, sin. ^. cos. ^. sin. ^. cos. (p+t^^'Sin. 'tf. cos. 'ip) + y (rffl. sin. i. cos. fl. cos. ^(p — dip. sin. *^. sin. (p. cos. ^) + a". rf«. sin. *e. cos. ip adding these quantities together and making the factors of the differentials of C, ^'i 9) which belong to the same coordinates coalesce we obtain — x".(dyp. COS. e-[-d<p)-\-2f'.(de. cos. ^ — rf'4'«sin. «. sin. <fi)s: 0 = (by substituting p and q instead of their values) x''. p — z''. y ; multiplying the first equation by COS. <5. the second by cos. 6. sin. ^, and the third by — sin. 6. sin. (p, we obtain dx'. COS. (p=jt".(rf\J/. cos. 6. sin. 9. cos. ^ — d(p. sin. (p. cos. ip) 4-y'((fiJ/. cos. «. cos. '^ — tfip. cos. ''ip)+z".rf4/. sin. 6. cos. <p oiy. cos. i- sin. ?i, = jc" (rf(p. cos. ^«. sin. (p. cos. ^ — rfl sin. t. cos, «. sin. ^<p — dy^. sin. 9. cos. <p. cos. ^) •\-tf'{d-^. sin. *<p. COS. ^ — d<p. cos. -d. sin. '<p — d6. sin. (). cos. 6, sin. ip. cos. ^) -j-s". rf«. COS. *«. sin. <p — dz!. sin. «.sin. tp=x".(d(. sin. d. cos. tf. sin. *^-\-d(p. sui. -tf. sin. (p. cos, ip) •\-y".(d6. sin. d. cos. 6. sin. <p. cos. (p — d(p. sin. '^ sin. 'ip)+a". rfd. sin. *d. sin. ^ adding and concinnating as before we obtain i/'.(d4'. cos. 6, — t/ip)4-«".(rf4'« sin. 6. cos. <p -{-</«. sin. ip)=0=: —j/'p +«".r 188 CELESTIAL MECHANICS, of y" and of £', angles of which the cosines are q r p 's/p' + q' + r* s/p^ + q^+r^ ' s/p'+g^+r' multiplying the second equation by sin. 6, and the third by cos. 6, we obtain dy. sin. (:=3/'.{d<p, cos. (p. sin. (i. cos. S—dl. sin. ^6. sin. (p — di^. sin. 6. cos. ^) •j-y.^d-^. sin. «. sin. (p — d<p. sin. 6. cos. 0. sin. i?>~rf«. sin. ^$. cos. ip)+^". £/«. sin. l. cos. I. rfi:'. cos. 6= — x".(d6. cos. ^6. sin. ?> -(- rfip. sin. $. cos. «. cos. ip) — i/'.{di. COS. ^^. cos. ip — rfip. sin. 6. cos. «. sin. ip) — i". c?«. sin. «. cos. 9 /. adding and concinnating we have — x".{d6. sin. (p + d^: sin. 1 cos. (p)—y".{dL cos. <>— ff-v}/. sin. «. sin. ?)= — x'V-f-y'. y. * The equations p/'—y2"—0—&c. are the equations of the projections of the line, relatively to which dx' dy' are equal to nothing at any instant, on the planes x" z" , y" je", &c. .*. the cosines of the angles which this line makes with the axes areVespectively For these cosines are equal to 9^' P I 1 I T ~ P P' and the same is true of the other cosines. From the preceding analysis it follows, that the locus of all the points whose velocity is nothing at any given moment is a right line, whose position with respect to the principal axes is determined by p, q, r, :. the preceding equations both evince the existence of such a line and indicate its position, and a body revolving about a fixed point may be considered as revolving about an axis determined in this manner, but as in general;;, q, r, vary from one instant to another, being functions of the time, the position of this axis will also vary, and hence it is that this axis has been termed by some authors the axis of instantaneous rotation ; whenp, q, r, are constant, the axis of rotation will remain immoveable during the motion of the system. PART I.— BOOK I. 189 Therefore this right line quiesces, and constitutes the real axis of rotation of the body. * * The values which have been given for px" — jz", pi/" — rz", qy" — rj", enables us to determine the linear velocity of each point resolved parallel to the axes of x' y and d for if we multiply the first of the preceding equations by cos. i. cos. (f. the second by cos. i. sin. f. and the third by sm. i. we shall obtain by adding them together — dJ . cos. i. sin, ip. cos. ip ■\-dy'. cos. '^i. cos. '<f — dz. sin. i. cos. 1 cos. '^^ ■^dx'. cos. «. sin. ip. cos. ip + f/y'. cos. ^^. sin. 2|> — rfz'. sin. i. cos. ^. sin. 2(p + ay. sin. ** 4-rfi'. sin. i. cos. «=:(// — (px" — ^z'). cos. «. cos. <P + {py" — rz"). cos. «. sin. ip -]-(qii/" — rx"), sin. « ; if we multiply px" — g^' by — sin. <p and ;;_/' — ra" by cos. ? we shall obtain (/c'. sin. -? — rfy. cos. e. sin. if>. cos. ip + rfs'. sin. 6. sin.(p. cos.ip..^«'' cos. *^ + rfy. cos. «. sin. (Ji. COS. (p-—dz. sin. d. sin. ip. cos. ip = dx' =—{px"— gz"). sin. ip+( pi/' — rz" ) . cos. ? ; multiplying px" — jaf ' by — sin. «. cos. 9, pi/' — rz" by — sin.fl.sin.ip. and qy" — rx" by cos.^, we shall obtain dx'. sin. 6. sin. <p. cos.^ — ay sin. 6. cos. S. cos. ^ ip -{-rfz'. sin. -*. cos. ^<p — cfx'. sin. j.sin. (p.cos. ^ — dt/.sin. t.cos.^.sin. ^<p + d^.sin. 'e.sin.*f-\-dy'.sin.e.cos.6. +rfa'.cos. " )=d!/— — (px"— jz".)sin. 6. cos.* — {py" — rzf'). sin. 6. sia.f + {qy" — rx").cos.« ; we might in like manner obtain the value of the accelerating forces resolved parallel to the axes of x' y' and z', by taking the differentials of dz', dy', dz', and of their respective values. Since as has been observed, in note, page 166, the coordinates of x'', y", z', do not vary with the time, and as the angles 6, -i^, ?, are functions of the time, it follows that when we take the differential of x"y'and z" respectively in terms of the coordinates i! ,i/, z', and of the angles 6, ■^, <p, the sines and cosines of these angles must be considered as constant, ,*. keeping this in view and also that sin. 4'=^0 after the differentiations we shall obtain dJ'z=.d3l. COB. <f-\-dy. cos. «. sin. <p — dyl. sin. «. sin. if ; d)f'-=i — dx' • sin. <f-\-dy, cos. i. cos. f — dz' , sin. «. cos. P; rfa"— d-tf . sin. «+rf/. cos. i ; 190 CELESTIAL MECHANICS, In order to determine the velocity of rotation of the body, let us consider that point of the axis of z''y of which the distance from the origin of the coordinates is represented by a quantity equal to unity. We shall have the velocities parallel to the axes of x of y' and of z', by making x" = 0, y" = 0, ;:"z= 1, in the preceding expressions of da^, dy\ dz', and then dividing them by dt, which gives for these partial velocities -r-, sm. 6 ; — . cos. 6 : ■, . sin. 6 ; dt ' dt dt therefore the entire velocity of the point in question, is y/d^* + cfvj/^.sin. *S di or \/^*+?'*, and dividing this expression by the distance of the point from the instantaneous axis of rotation, we shall have the angular velocity of rotation of the body ; but this distance is evidently equal to the sine of the angle, which the real axis of rotation makes with the P axis of z", and the cosine of this angle is equal to — — ; / but it is evident from what precedes that the second members of these equations are equal respectively to we have dx" ill/' rfz" consequently the quantities p, q, r, which determine the position of the axes of rotation, give also for any other point the linear and angular velocities of the different points of the body resolved parallel to the coordinates x", y", and z". PART I.— BOOK I. 191 therefore s/p*-{-q*-\-r* will be equal to the angular velocity of rotation. * It appears from what precedes that whatever may be the rotatory motion of a body, about a fixed point, or a point considered as fixed ; this motion must be considered as a motion of rotation about a fixed axis during an instant, but which may vary from one moment to another. The position of this axis with respect to the principal axes, and the angular velocity of rotation depend on the variables p, q, r, the de- termination of which is most important in these investigations, and as they express quantities independent of the situation of the plane of x' and y, are themselves independent of this situation. 29. Let us proceed to determine these variables in functions of the time, in that case in which the body is not solicited by any accelerating forces. For this purpose, let us resume the equations (Z)) of No. 26, existing between the variables p', q, r, which are in a given ratio to = the cosine of the angle which the axis of a" makes with the instantaneous axis of rotation .*. p« yz+r- is equal to the square of the sine of the same angle, and since i" is by hypothesis equal to unity we have the perpendicular distance of the-point in question from the axis of rotation equal to this sine, .-. dividing \/ (f+r^ by this distance the quote will be equal to 's/ j^-\-q^^r^, and as the axisduiing an instant may be considered as fixed the angular velocity of all points during this instant will be the same, the selection of the point so circumstanced that x"=iO,^'=0, 2"=il is made in order to simplify the calculus, .-. it appears from an inspection of the value of the angular velocity, that it is constant when p q and r are constant, i, e, when the axis of rotation is immoveable, but the converse of this proposition is not true for it is possible that the function ^ pii^q^i^s si,ould be con- stant, while at the same time its component parts may vary, see page 197. 192 CELESTIAL MECHANICS, the variables p, q, r,.* In this case, the differentials dN, dN', dN" waxiish, and tliese equations being multiplied by ;/, </', and r' respectively and then added together give 0=p'.dp' + q'.dq' + r'.dr' j and integrating them we shall obtain k being a constant arbitrary quantity. If we multiply the equations (Z)) by A B.p, BC.q, and AC.r', and then add them together, we shall obtain by integrating their sum, AB.f-^ BC.q'"' + AC.r'- =. H"- ; /Z being a constant arbitrary quantity ; this equation involves the prin- ciple of the conservation of living forces, t By means of the two pre- * p: p' :: 1 : C :: 1: S{s"^+i/"%dm, but this is a constant ratio, because the position of the principal axes being given, the quantity S{x"^-\-y"^). dm is constant, and when no exterior forces act on the body, the quantities N, N', N", are constant and /. dN dN' dN" vanish. -]- For substituting for p', /, /, their values, we obtain A.B.C(Cp^+Aqi + Br')-m,.:S{x"^+y"-)dm.p'-+S(y"'-i-z"')dm".q^ + S(x" + s"M(/m. r'= a constant quantity, now we have seen in a preceding note, that the velocity of any point resolved parallel to the axes of Jt" of ^'' and of sf' is equal tof «"— js", pf—r~J', qy"—rJ' and the sum of the squares of these quantities the square of the velocity of the point whose coordinates are x", tj' , z", .: this expression multiplied by dm equals the living force of this molecule, now as the quantities p, q, r, are the same for all molecules at the same instant, the sum of the Uving forces of all the PART L— BOOK I. 193 ceding integrals we shall obtain .^_ AC.k"—H'+A.(B—C). p' ^ ^ - C. {A—B) '2_ i/ — BC.Ir-B.(A—C).p" ^ ~ C(A—B) thus, we shall have q' and r in functions of the time, when p' will be determined, but from the first of the equations (D) we have ^^ {A—B).q'r. ' consequently c c molecules will be equal to P' •. A" ' +y" ' )-'^»« + 9 Y(y" * +z"f.dm-{-r.lfx"^+z''s)^m —2pg/x"z".dm—2prfi/'s^'.dm—2qrfi/':^'.dm, but these latter quantities vanish, x",t/", z", belonging to the principal axes consequently j^f{x"^+y"%dm +q-/ij"^4-z9^)dm+t^f{x''^+z"^).dnt is equal to the sum of the living forces, and being constant as has been just shewn, it fol- lows that the expression AB.p'^+BC. q'^+AC. /'~H% involves the principal of the conservation of living forces. ^ 'o ,« « , m—AB.j/^—BC.g'^ * r^=:P—j/'—g'i= ■—■ i— .-. AC£—ACf-—AC.c/^ = H^—AB.p'-—BC. /^ therefore g'' = AC.k-—H^+ A.{B—C)p ^ C.(A—B) the value of /^ is derived in a similar manner. 194 CELESTIAL MECHANICS, ABC.dp' dt= s/ \AC.Ic^H? 4- ^.(ii— C').y'i . {H'—BC./c'—B.(A—C).p"] * * When A-B, dt= , ^^^■"'PJ ,y{AC.Jr—H-+ A\A—C)p''). (H-—AC/c'—A(A—C}.j/*) ABC.dp' ^ ABC. dp' A Ck''—tr^+A{ A—V).p'^ and it may be made to assume the form a-- dp' , ^ ,_.^^ , ^ ABC ACh^—H-" A.{A—C) and a' being equal to - and the integral of this expression = t=C^. (arc tangent=y to radius= a, the constant quantity is equal to nothing because t=0 at the same time with/>'. When A=.C the expression for dt becomes AB.Cdp' ^{A- k-—H- +A.{B—A)iP). {H''—BAIr) this expression may be reduced to the form C,- — 1 — ^^ ( in which C, is equal to - \ ' ^1F+^^ ' ^ A^B-A).{m-BA.k ) Ak^—H2 and a-=- A.{B—A) the integi-al = C,. log (/+ ^/ a2+;.« If2?=Cthenc/^. ^^'^P' _c.—±^PL '/(ACJ.^—H'^j(W—B-k'—Bi(A—B)).p^ ' s^ a^—p'* and the integral will be arc sine =p' rad = a ^, ^ . , AB' C, Dem£; equal to " / • ' ^ ^ <^AC/c^—hK(H^—B''k^) and a^ = iJCIf-H% jlP-m^ —B.(A—B) if /lC.4* = fl2theo PART I.— BOOK I. 195 this equation is only integrable in one of the three following cases, B=A, B=C, A=C. The determination of the three quantities p', q, r', involves three arbitraiy quantities, H.\ Ji^ and that which the integration of the pre- ceding differential equation introduces. But these quantities only give the position of the instantaneous axis of rotation of the body, on the surface, /", e, with respect to the three principal axes, and its angular velocity of rotation. In order to have the i-eal motion of the body, about the fixed point, we must also know the position of the principal axes in space ; * this should introduce three new arbitrary quantities c c 2 ,o A.{B—C).p'2 ,^ ABC dp' ^ t{A—B) ^ A(,B—C)if\H'i—BClr—B.[A-C)iy- ^adv = c,:t - — in which 2C - AEC V (/i{B—C) H'— BC. /c^^ ~ —B,(A—C) its integral will be equal to C,, log. ^ v " —p a + sja^ — i)" See Lacroix, page 256, No. 174.. and if m=BC.lc^ then dt^ -'^^•^/'' \/ ACIfi—H^+A{B~C)pfi)[—B{A—C)p'^) ^^ and t= C. W. '^'^W^a =C,. and t~ C,. log. the constant quantities vanish for these integrals, because as has been already mentioned p'=0 when t vanishes. The value of dt cannot be exhibited in a finite foim except in the cases already specified, and when all the moments of inertia are equal, in every other case, the value of the integral of dt must be obtained by the method of quadratunes. * From the quantities/, g', r', we can collect the values of p, q, r, which are in a given ratio to them, and from these last quantities we obtain the cosines of the angles whicli the axis of instantaneous rotation makes with the principal axes, but as these axes though fixed in the body are moveable in space, we must know the position of these axes at the com- 196 CELESTIAL MECHANICS, which depend on the initial position of these axes, and which i-eqviire three new integrals, which being joined to the preceding quantities will niencement of the motion, in order to have the real motion of the body, which gives three constant quantities. Substituting in the values o{—N^ — N', — N", p' for Cp, q for Aq, r' for Br we shall have q', sin. 6. sin, (p + r. sin. «. cos. <p — p. cos. 6, ~ — A'^ q'. COS. $. sin. cp. cos. ■^■\-r'. cos. 6. cos. (p. cos. -^ '\"p' • sin- ^- cos. -i^ •\-r'. sin p. sin. 4^ — q. cos.ip. sin. 4^ — iV' — 9'. cos. £. sin. If. sin.i|/ — ?'. cos J. sin. 4". cos. <f> — ^'. sin. ^. sin. 4- -}-r . sin. (p. cos. ■v}'^?'- cos. <p. cos.-v^ :;: — W squaiing these quantities we obtain 9'. Vsin. ^«. sin. ^<p^r'? sin. *«. cos.'ip + p'.^cos. 'e-\-2q'r'. sin. '«, sin. 1?. cos. ? — Sp'y'. sin. ^. cos. S. sin. (f — 2p'r'. sin. 6. cos, ^. cos. <p=N^ q'.'cos. *^.sin. »(p cos. ^-^-^r'.^ cos.'<. cos. 'ip.cos. ^4'-\-p'' ^ sin. '^.cos. ^t^ + 2//. cos. -J. sin. (p. cos. ?>. cos. '4/+2/>V'. sin. S. cos. ^. cos. <fi. cos. ~--^ + 2;/^'. sin. <. cos. 6- sin. (p. cos. *4) -}-r'.- sin. -?i.sin. 'ij'-l-/-' cos. ^ip.sin. ^■^ — 2/,/, sin. <p. cos. ip. sin, ■^=N'^ j'.* cos. '6. sin. *{?. sin. 'tJ/^/.s cos. '^. sin, '■J'- cos, '0-j"P'-' ^'f- '^- ®'"- '^ -j-Sg'/. COS. --«. sin. ' 4^. sin. (f. cos. <t>-^-2p'r' sin. S. cos. «. sin. 'if'' cos. (p -^2p'q' sin. <. COS. S. sin. ^■^. sin. if) •j-/.* sin. "ip. COS. '4'+o'. ' COS. ^$. cos. '4' — 2yV. sin. (p. cos. ip cos. *4=-^"" /. adding the first members of these equations together we obtain q'.- sin.'*, sin. •tp-^-q','^ cos. ^6. sin. ^ip+y'.^ cos. '?>=(y'^ )-{-/.» sin, '^. cos. *? 4- r'.^ COS. '^. cos. 2(p + /,» sin. '(?= (}•'*) +j3'.' cos. '6-\-p'.^ sin. *«. cos. '4^ -{■p'.' sin. «^, sin, ^4'=/'';* the parts of these squares which are the products of two different quantities vanish when added together and in the expressions for A^',* A'*",* we omit the product {q'. cos. 0. sin. p cos.4' + »•' COS. (. cos, ip, cos. 4' •{•p'- sin. i. cos. 4')'('''' sin. (p. sin. 4'— ?• cos, ^. sin. 4^) for this pro, d,uct occurs in A^" and Nff* affected with contrary signs, .•, it must vanish from A^'2+ A"'* .*. we shall have p'*+q'*+r'^ Zi + N'+N'^+N"^. * PART L— BOOK I. 197 completely solve the problem. The equations ( C) of No. 26, involve three arbitrary quantities N, N', N", ; but they are not entirely distinct from the arbitrary quantities H and k. In fact, if we add together Ihe squares of the first members of the equations (C), we shall have ;/' + f 4- r'"- =N"-+N" + N" ' ; and consequently The constant quantities N, N', N", correspond to the constant quan- tities c, c, c", of No. 21, and the function t. t. \/p'^+q'~-\-r- expresses the sum of the areas described in the time t, by the projection of each molecule of the body on the plane relatively to which this sum is a maximum. iV', N", vanish with respect to this plane, .•. if we put their values, which have been found in No. 26, equal to nothing we shall have 0 = Br. sin. (p — Aq. cos. f ; 0—Aq. cos. 9. sin. (p + Br. cos, 9, cos. (p+Cp. sin. 9 ;* ♦ From the equation Bi: sin. ip—Aq. cos. ^ rr 0 we obtain by substitution tan. (p = ~ .-. COS. <p. = — ::iz^:iir ^n" sin. <p= ::::£^:^r « consequently we have ' — . cos. (, "^•v^ ill'^+r'^)- COS. l=p'. sin. e..: {q'^ +r'^). cos. '^6=p'^— p.- cos. 't .: P COS. i = •v/ ?"+?'*+»■' if we multiply the first of the preceding equations by cos. 6. sin. (p. and the second by cos. <p. we shall obtain by adding them together /. cos. O+p'. sin. 6, cos. ^s=0 .'. substituting for cos. 6 its value we obtain 198 CELESTIAL MECHANICS, from which we deduce COS. B =- >//H!7'^+r' sin, 6. sin. 9= ■ .-„^ . •:= > Vp'+^'+r"'' sin. V. COS. 0 = By means of these equations, we can determine the values of S and <p in functions of the time with respect to the fixed plane which we have considered. We have only now to determine the angle 4', which the in- tersection of this plane, and that of the two first principal axes, con- stitutes with the axis of a:'; but this requires a new integration. From the values of q and of r which have been given in No. 26 we derive e?i)/. sin. '■6==q.dt. sin. 9. sin. (p + r.dt. sin, 6. cos. p ; from which we deduce — / sin. ^. COS. ®= and if we multiply the first of the preceding equations by cos. 6. cos. (p, and the second by sin. 1^ and then substract the first fi-om the second we shall obtain q'. cos. 6+p'. sin. 6. sin. <p = 0 .'. substituting for cos. I its value, we obtain sin. 6. sm. 9= \/p"-+'f + r"^ PART I.— BOOK I. 199 _ —k.dt.(Bq' '-\-Ar'^)^ ^~ AB.iq^ + r"') but from what precedes, we have H'—AB.p''- q'-+r'i=k^—p"', Bq"-^Ar'^= ^ i- j therefore we shall have _ —k.dt(H'—AB.p'-) ^'^- ABC [k^—p') By substituting in place of dt, its value which has been given above ; we shall have the value of »J> in a function of p' ; thus the three angles e, ip, and ip will be determined in functions of the variables^', q', r', which will be themselves determined in functions of the time ^.t Consequently we can have at any instant the values of these angles with respect to the plane of x', and y, which we have considered, and it will be easy by means of the formulae of spherical trigonometry, to » If we multiply the values of qdt, rdt, given in page 166, respectively by sin. (. sin. f, sin. i. COS. <p, and then add them together we shall have d-^. sin. 2^ = q.dt.sm. (. sin. <p-\-rdt. sin. 6, cos. if = \ A,k B.k AB,k k^ the value of d^' will be f Cos. «— sin. e. sin. <;>, — sin. 6. cos. p, are the cosines of the angles which a perpendi- cular to the fixed plane or the axis of z' makes the principal axes, see page 180, and P' 9' -^ are the cosines of the angles which the principal axes, a''-', i/", x", make with the axis of the plane, on v/hich the projection of the area is a maximum, consequently the cosine of the angle wliicli the axis of the plane on which the projection of the area is a maximum makes 200 CELESTIAL MECHANICS, . * determine the values of the same angles with respect to any other plane this will introduce two new arbitrary quantities, which combined with the four preceding quantities will constitute the six arbitrary quantities, which ought to give the complete solution of the problem which we have discussed. But it is evident that the consideration of the above men- tioned plane simplifies considerably this problem. The position of the three principal axes on the surface, being supposed to be known ; if at any instant, the position of the real axis of rotation on this surface, is given and also the angular velocity of rotation, we with the axis of the fixed plane, (see note to page 7). ■p' . cos, 6 — /. sin. 6. sin. 0 — ?■'. sin. 6. cos.ip N, we might by a similar process shew that the cosine of the angle which the axis of the plane of greatest projection makes with i/', and x', are respectively proportional to N' and A'''', consequently the position of this plane with respect to the fixed axes oi x' y', and z is given, therefore this plane remains fixed during the motion, and the values of N, N', N'', are the three quantities which determine the position of the fixed axes, with respect to the plane of greatest projection. , * The determination of f^,q'y, which give the position of the instantaneous axis of rotation requires three arbitrary quantities and (he determination of 6, ■^, (p, which give the position of the principal axes with respect to the fixed axes requires three more arbitrary quantities, these are, H, k, and the constant quantities which are introduced by the integration of dt and d-^, the two remaining quantities are determined by the values of cos, 6, sin S. sin. <p, sin. 6. cos. (p, for any other fixed plane beside the invariable plane, .'.by making the plane of greatest projection, to coincide with the fixed plane ; these new arbitrary quantities vanish, and the number of constant arbitrary quantities will be reduced to four. The values of 6, (f, ^, with respect to the plane on which the projection of the area is a maximum being given, and also the value of the angle which this plane makes with any other plane, it will be easy to deduce the cosine of the angle which each of the principal axes makes with the assumed plane, in fact by means of the values of N N' N" we can de- termine the angles (, ^, (p, where we have the values of the same angles for the plane on which the projection of the area is a maximum, i, e, where we have^, q', r', and substituting p, q, r, in place of p', if, /, in these expressions we obtain the cosine of the angle which the axis of instantaneous rotation makes with the axis of the fixed plane, the three quan- tities A', N', N", are not undetermined, for if N' and A'* have definite values the value of A^ is determined by means of the equation A'-(- A^'-f- A**=^. PART I.— BOOK I. 201 shall have the values of p, q, r, at this instant because these values di- vided by the angular velocity of rotation express the cosines of the angles, which the real axis of rotation constitutes with the three prin- cipal axes ; .*. we shall have the values of p, q, r', but these last values are proportional to the sines of the angles which the three principal axes constitute with the plane a/ and y', relatively to which the sum of the areas of the projections of the molecules of the body, multiplied respectively by these molecules, is a maximum ; therefore we can determine at all instants, the intersection of the surface of the body with the invariable plane ; and consequently find the position of this plane, by the actual conditions of the motion of the body. Let us suppose, that the motion of rotation of the body arises from a primitive impulse, of which the direction does not pass through its centre of gravity. It follows from what has been demonstrated in Nos. 20 and 22, that the centre of gravity will acquire the same motion, as if this impulse was immediately applied to it, and that the body will move round this centre with the same rotatory motion as if this centre quiesced. The sum of the areas described about this point, by the radius vector of each molecule projected on a fixed plane, and multiplied respectively by these molecules will be proportional to the moment of the principal force projected on the same plane j but this moment is evidently the greatest possible for the plane which passes through its direction and through the centre of gravity ; consequently this plane is the invariable plane. If the distance of the primitive impulse from the centre of gravity be^and if w be the velocity which is impressed on this point, m re- presenting the mass of the body, mfv * will be the moment of this im- D D • V being the velocity of the centse of gravity, and m being the mass of the body, the measure of the force will be equal to mv, and its moment with respect to the centre of gravity will be equal to inf.v, see No. 3, and the motion of all the molecules of the body arising solely from this impulse it is evident from the principle of D'Alembert, which has been established in No. 18, that the quantities of motion which these molecules have at the commencement of the motion, estimated in a direction contrary to their true direction must 202 CELESTIAL MECHANICS, pulse and being multiplied by iJ, the product will be equal to the sum of the areas described in the time /, but by what precedes this sum is equal to — . ^j>^-\-f^r'" ; consequently we have If at the commencement of the motion we know the position of the principal axes with respect to the invariable plane, i, e, * the angles 6 and (p ; we shall have at this commencement the values of p' q and r and consequently those of /?, q, r, therefore at ani/ instant we shall have the values of the same quantities, t CQUStitvite an equilibrium with the force mv consequently the principal plane i, e, the plane with respect to which the moment is a maximum is the plane passing through tlie centre of gravity, and the direction of the primitive impulsion .•. the sum of the areas described in the timef =\.t.mfv. * The constant quantity k ^ m.fv ; in order to determine H, it may be remarked that the position of the principal axes at the commencement of the motion, with respect to the pkme passing through the fixed point and the direction of the impulse being given, we have the the values of f q, r, being proportional to the cosines of the angles which the principal axes make with the axis to the invariable plane. Consequently we have the constant quantity the third constant quantity will be determined by integrating the value of dt, which will be equal to a function of p'-\- a constant arbitrary quantity ; // which is proportional to the cosine of the angle which the axis of 3" makes with the axis to the plane of greatest moment lias a detemiined value when <=;0 . • . by means of this value we are enabled to find the value of the third constant quantity ; with respect to the fourth constant quantity which arises from the integration of the value of d-^, this gives il' = to a function of p plus a constant quantity, p' being proportional to cos. (, we shall obtain the fonrth constant quantity which is necessary to complete the solution of the problem, if we know what value of 4' cor- responds to a given value of i. f \Mien a solid" body is not eolicited by any accelerating forces and can revolve freely about a point we shall have dx.zs.yi'S'^^i^ydyzi.id-^—Kdin, dxp=,sd!pr-yd^, &c. PART I.— BOOK I. 203 By means of ttis theory, we are enabled to explain the double motion of rotation and of revolution, of the planets, by one initial impulse. In fact, let us suppose that a planet is an homogenous sphere whose radius is D D 2 See page 89, if we multiply the equations (^Z) of No. 21, by rfw d^ d-^ ^' df' W respectively we shall obtain d-a , d<p ,, dyl' C xdT^.diz—y.d'^a.dx ? , „ C zdip.dx — xd(p.dz 1 '■dr+'-^+''^^^"'-i — -w 5 +'• i — df' 5 ^^^^ ^y.d4^.d.-..dMy,-^ ^,„,, {^-^— }-d^ + ^^» {'-^^^}.d. ' (y.d-^—xdtp) , ^ {d3^-{-dyi+dz'i) const, (see No. 19) now if we substitute fof c, d, c", ^cS+c/^+c".^ COS. 6, ,yc^+d^ + c".« sin. «. sin. ■f, — ^/^+c'!'+ c".!' sin. ». cos. i^, to which they are respectively equal, and also for dvr, dip, d-<p, ds. COS. I, de. cos. n, ds. cos. m, see page 90, we shaU obtain (cos. i. COS. ^-(-sin. ^. sin, i^. cos. n — sin. i. cOS. ■^. coS; m) :Z const, as cos. 6, sin. 9. sin. 4', sin. 6. cos. t^i are equal to the cosines of the angles which the axis of the plane of greatest projection, makes with three fixed axes, and as cos. I, cos. n, cos. m, are the cosines of the angles which the axis of instantaneous rotation makes with the same axes, the last factor of the second member of the equation is equal to the cosine of the angle, which the axis of rotation makes with the axis of the plane on which projection of the areas is the greatest possible, . ■. as di . . - — is the exponent of the velocity of rotation for any instant, this expression multiplied 204 CELESTIAL MECHANICS, equal to i2, and that it revolves about the sun with an angular velocity equal to Z7; r being supposed to express its distance from the sun, we shall have t;=r U; moreover if we conceive that the planet is put in motion by a primitive impulse, of which the direction is distant from its centre by a quantity equal to j^ it is evident that it will revolve about an axis pei-pendicular to the invariable plane ; therefore if we suppose that this axis coincides with the third principal axis * we shall have 6=0j and consequently (/'=0, r' = 0; therefore /?' = ««^ ?, e, CjizzmfrU, But in the sphere, we have C ■=: — mR* ; consequently, o f- ^E p. which gives the distance of the direction of the primitive impulsion from the centre of the planet, and satisfies the ratio which is observed to obtain between p the angular velocity of rotation, and U the angular velocity of the revolution of the planet round the sun. With respect to the earth, we have ^= 366,25638 ; the parallax of the sun gives — ::;: 0.000042665, and consequently y = — . R very nearly. into the cosine of the angle, which the axis of instantaneous rotation makes with the axis of the plane, on which the projection is a maximum, is a constant quantity. When the plane oi x ij coincides with the plane passing through the direction of the impulse, and the point about dp d^' which the rotation is performed cos.«=l and sin. e = Q .: we shall have c' -j-, c"— — 0 ; di constant quantity=c. — . cos. I consequently the velocity of rotation, i. e, parallel to the axis of 2,= -r-. COS. f IS constant. at * All the diameters of a sphere being principal axes, if we suppose that the axis of revolu- tion wliich is evidently the axis of the invariable plane coincides with the axis of y", * = 0 .-. cos. fl=l .-. q' and / = respectively to sin. i. sin. ip, sin. 6. cos. <p vanish and this con- siderably simplifies the calculus. PART I.— BOOK I. 205 The planets are not homogenous ; but we may suppose them to be com- posed of concentrical spherical strata of unequal density. Let /> denote the density of one of those stratas of which the radius is equal to R, we shall have ^_ 2ot fp.RMR , 3 Jp.R.-dR ' * The moment of inertia for a sphere is calculated in Book V. No. II. in a general manner, but as it involves some steps which are demonstrated in the second and third books, it will be necessary to give here a special demonstration, let there be two concentrical circles, whose radii are q, q-\-dq, the circumference of the interior is equal to '2-jr.q, and the area of the annulus contained between the peripheries of those circles is equal 2v,q.dq .'. 2%.q?dq is equal to the moment of inertia of this annulus and l^.q,* is the moment of inertia of a concentrical annulus of a finite breadth, .*. when the preceding integral is taken between the limits y=0, q = R the expression becomes ^ttR*, which is the moment of inertia for the entire circle, now in order to obtain the moment of inertia for the entire sphere, let us conceive a plane parallel to the axis of rotation cutting the sphere at a distance from the axis equal to x, its intersection with the surface of the sphere will a lesser circle of the sphere, let ^=: the radius of this circle, the moment of inertia of this circle with respect to its centre is equal by what precedes to ^ vy* ,•. the moment of inertia of an indefinitely small slice is equal to ^■!r.y*.dxz:i Itt (2Rx-~x*)- .dx, for i/^zz2Rx—a:^ R being the radius of the sphere, .•. btegrating we have -^^>-Wro} — the moment of inertia of a spherical segment and this integral being taken between the limits_a;=0, and x=R gives — the moment of inertia of the entire sphere with respect to a diameter, and it is very easy by means of the expression which has been given in page 180, to obtain the moment of inertia for any axis parallel to the diameter, if R is supposed to be variable in the last expression, and if § the density varies from the centre to the circumference, the moment of inertia of any spherical stratum whose radius== R is ^.t^RUR 10 206 CELESTIAL MECHANICS. (p being a function of R). If, as is very probable, the denser strata are nearer to the centre j the function /' p'g ,„■ will be less than , consequently the value off jf' Jx, uK 5 will be less than in the case of homogeneity. 30. Let us now determine the oscillations of a body when it turns very nearly, about the third principal axis. We might deduce them from the integrals which we obtained in the preceding number ; but it is .-. the moment of inertia of a sphwe composed of concentrical strata is equal to g- Ty g./Z* dR, m like manner m=the mass of the sphere =i4tir./^R*dh m am d f~ ^'P 8w pf^R*dR _ Jp.f^.R*.dR_ Vs-R*'^^"" 7n.TU ^3Atn.Tlir^.E'dR'-' 3rUfi,R^.dR we obtain the ratio of U to p from knowing the period of the earth and the time of its rotation, for the angular velocities are mversely as the angles described in the same time, 5 being by hypothesis a function of R where the density increases towards the centre e^ — =- .•. the fraction in the text becomes R\dR <P.(R) ^ ^ R^dR i^R) by parUal integration _Rfi_ R5A.q>[R) Rs R3.d<p{R) 5.<pR ^ 5WR) f "^ 3(p{R) •' S.{(pRr and as the numerator is more diminished than the denominator the value of the fraction 5 SR* which in the case of homogeneity was -3— will be diminished when the density mcreases towards the centre. PART I.— BOOK I, 207 simpler to deduce them directly from the di£FerentiaI equations (D) of No. 26. The body not being actuated by any forces ; these equations will become by substituting Cp, Ag, and Br, in place of their respective values p', q', r. dq + ^^^.rp.dt^Oi rfr+ ^—Sl.pq.dtzzO, The solid being supposed to revolve very nearly about the third prin- cipal axis, q and r * are very small quantities, therefore we may reject their squares and products ; consequently we shall have dp^^o and p will be constant. If in the other two equations we suppose q^M. sin. (w?+v) } r=3=M'. cos.(n?+y) ; we shall have * The solid being supposed to revolve very nearly about the principal axis, the cosine of the angle which the instantaneous axis of rotation, make with the principal will be q.p, equal to unity consequently, j and r will be very small because the sine of the above mentioned angle which is equal to \/£+r very nearly vanishes. 208 CELESTIAL MECHANICS, „=,. / CC-..).(C-.) ., ^._ _ mV ggM ana ,• being two constant quantities, the velocity of rotation will be y/'f-^q^-Yr' or simply p, the squares of q and r being neglected j therefore this velocity will be very nearly constant, finally the sine of the angle formed by the real axis of rotation, and the third principal axis will be y/g' + r V If at the commencement of the motion we have ^=0 and r'=0, i, e, if at this instant the real axis of rotation coincides with the third prin- cipal axis ; we shall have M =0 M'= 0 ; consequently q and r will be always equal to nothing, and the axis of rotation will always coincide with the third principal axis ; from which it follows that if the body commences to revolve round one of its principal axes, it will continue to revolve uniformly about the same axis. It is from this remarkable pro- * q=.M. sin. (ni+y) r = M' . cos. (nt+y) satisfy the preceding differential equations, for by substituting these values we obtain MmAI. (cos. (n<+y)+ —r-^-V M'-^^- COS- (nt + y) ZZ 0 ^ (J — M'.n.dt. sin. (nt+y)-\ — ■^—.pM.dt. sin. (wf+y)=0 .-. Mn+^^~^^ .pM'=0—M'.n+^t^.pM=0 .'. M'=: Mn.A _ (A—C).p.M ,_ '~'p.{C—B)~ Bn •'•" — (C-^)(C-i^). 3,,^ _^. .1 A. jC-A) . ^ AB V B.{C—B) the quantities M and y are arbitrary consequently these values are perfect integrals of the two preceding differential equations which they satisfy. (See Lacroix traite elementaire de Calcul integral, No. 297). PART I.— BOOK I. 209 perty, that these axes have been termed principal axes of rotation, it ap- pertains to them exchisively ; for if the real axis of rotation is in- variable on the surface of the body, we have c?p= 0, dq=^ O, dr=- 0, there- fore from the preceding values of those quantities we obtain (5—^) ^ {C—B) ^ (A—C) ^^—^—^-rq =0 ; ^_ — _.r/J=0 ; )^—L.pq= 0. In the general case where A, B, C, are unequal, two of the three quantities j3, q, r, vanish in consequence of these equations, which implies that the real axis of rotation coincides with one of the principal axes.* If two of the three quantities A, B, C, are equal, for example, if we have A=B ; the three preceding equations will be reduced to the follow- ing, r/j=0, pq=0 ; and they may be satisfied by supposing^? =0. The axis of rotation in this case exists in a plane perpendicular to the third principal axis ; but we have seen in No. 27, that all axes existing in this plane, are in this case principal axes. £ E * The value of the quantities M, M', may be determined by knowing the position of the instantaneous axis of rotation at the commencement of the motion, whatever be their values at that instant they remain unaltered during the motion of the body .'. if at the commencement of the motion, the real axis of rotation coincided with the principal axis .-. 5 and rare respectively equal to nothing, and therefore M and M! will vanish, conse- quently the values of q and r will always be equal to nothing, and as p is constant and equal to the angular velocity, the body will revolve uniformly about the principal axis. If the position of the real axis of rotation is invariable on the surface of the body, p, q, r, must be constant, see No. 29, page 201, .•. rf/j, dq, dr, are respectively equal to nothing .-. their values B—A C—B A—C respectively vanish, .•. in order to satisfy these equations two of the three variable quantities p, q, r, must vanish. 210 CELESTIAL MECHANICS, Finally, if the three quantities j4, B, C, are equal, the preceding equations will be satisfied, whatever may be the values of p, q, r ^ but in this case, all the axes of the body are principal axes.i* It follows from what precedes, that to the principal axes only belongs the property of being permanent axes of rotation ; t but they do not *Wlien ^=5, the first of these three equations vanishes of itself, whatever maybe the values of r and q, and we shall satisfy tlie two last equations by supposing p^O, .: the real axis of rotation is perpendicular to the third principal axis, see No, 29, notes, but as in this case all lines drawn in a plane perpendicular to the third principal axis, are principal axes, it follows that the axis of rotation is in this case a principal axis ; if A:=.B~ C the three pre- ceding equations will be identical, and the values of p, q, and r, may be assumed at pleasure, hut in this case all axes are principal axes, .*. it follows universally, that if the axis of rota- tion remain permanently the same, it must be a principal axis. In the general case when A B and C, are unequal, we shall be always certain that p, q, and r, and M, M', vanish at the commencement of the motion, when the impulse is made in a plane which coincides with the plane of two of the principal axes, for in this case the invariable plane to which we adverted in Note to page 184, coincides with the plane passing through two of the principal axes, and the axis of rotation or of this invariable plane will necessarily coincide with the third principal axis. See Notes, to page 188. ■f It might be proved directly from the property of principal axes scilicet S^z.dm ■= 0, Syz.drtf^ 0, that the pressure on the axis of rotation wliich is produced by the centrifugal force must vanish, when this axis is a principal axis, and that consequently, when there is a fixed point given in a body, there exists always three axes passing through this point, about which the body may revolve uniformly without a displacement of the axis, and as if these lines were entirely free ; for if the body is acted upon by an initial impulse, •n- denoting the angular velocity and r the distance of a molecule dm fi-om the axis of rotation wliich we suppose to coincide with the axis ;;:, x and y being the coorilinates with respect to the axes of X and y, we liave the centrifugal force^w'r.rfm, this force resolved parallel to x and y:z. '■ — .dm, — ^, because —,— are equal to the cosines of the angles which the axes of x r r r r ■ and^ make'with r, ••• the sum of the forces for all the molecules of the body = •a-^Sx.dm, ■a^.Sydm, and the respective sums of their moments for the axes of y and of x are •sr* Sx.z. dm, v?. S.yz.dm. and m being the mass of the body and x^, y,, being the coordinates of the centre of gravity, wehave •a'^ .mx^^-a^SJcdm, ■c:;- .niyj='S!^S.y.dm, and if z^z^, repre- sent the distances of the resultants w^-mx, ar*wzy,, from the plane of the axis of x^ we have by note to No. 3, ■B^inx^,^'a^SxyJm,TT '^ .myz^z=-a'^ .Syz.dm,vfheB z,z,i are equal, the resultants PART I.—BOOK I. 211 possess this property in the same manner. The motion of rotation aboirt the axis, of which the moment of inertia is intermediate between the moments of inertia of the two other axes, may be disturbed in a sensible degree by the slightest cause ; so that in this motion, there is no stability. The state of a system of bodies is termed stable, when the system being very slightly deranged, it deviates from the state by an indefi- nitely small degree, by making continual oscillations about this state. This being understood, let us suppose that the real axis of rotation deviates from the third principal axis by an indefinitely small quantity ; in this case, the quantities M and M' Si\e. indefinitely small ; and if n is a real quantity, the values of q and r will always remain indefinitely small, and the real axis of rotation will only make excursions of the E E 2 m^.mxp-a^.myi, are applied to the same point, .'.these two forces will compose one sole force =^z!r«. m \/.r/-|-?//, now if the fixed axes pass through the centre of gravity we have x==Oy,=zO .: 2=-^ Sjcdyn, ct ■^.Sydm respectively vanish, and if the axis of rotation is a principal axes we havear2S.xz.rfm=0, ^'^ S.yzdm=iO, from the first equation it follows that the axes does not experience any tendency to a progressive motion, and the second equations indicate that the sum ■a ■" . Sxzdm of the moments of the forces vanish, from these two conditions it follows that the forces constitute an equilibrium independently of the axis. If the fixed axis of rotation and origin of the coordinates was transferred to a different point of the body, being still a principal axis, we should have as before S.xs.dm = 0 Syz.dm =0 ••. the sum of the moments of the forces with respect to the axes of y and of x vanish as before .*. as Xj and y^ have in this case a finite value z, and z^, must vanish, for id ^ .mj,, z,, na^.my^ z^^, vanish being equal to ■ot'. S.xzdm, m^-Syzdm, ,; the pressure = w*.m v V*+^'*., which as z, z„ vanish, must exist in the plane of x, y, and must pass through the origin of the coordinates, . • . if this point is fixed the pressure will be destroyed, and the motion will be performed about the axis as if it was fixed, for the only pressure which could displace it is destroyed, by the resistance of the fixed point. From what precedes it appears, that when the principal axis passes through the centre of gravity, it is not necessary that any point should be fixed, in order that the motion may be perpetuated uniformly about the fixed axes, in any other case it is necessary that the origin of the coordinates be fixed. 212 CELESTIAL MECHANICS, same order * about the third principal axis. But if n was imaginary, sin. (wf+y), cos. (nt+y) will become exponential, and the expressions for q and r might then increase indefinitely, and at length cease to be very small ; consequently there would be no stability in the motion of rotation of the body about the third principal axis. The value of n is real, if C is the greatest or the least of the three quantities A, B, €; for the product (C — A). (C — B) is positive ; but this product is negative if C is intermediate between A and J5, and in this case n is imaginary ; thus, the motion of rotation is stable about the two principal * When n is a real quantity, p and q can be expressed by sines and cosines of nt, but these values are not susceptible of indefinite increase with the time, for they are periodic functions of t, and the limit of the values of sin. {nt-\-y), cos. {nt-\-y) is unity, if they are very small at the commencement of the motion, M and M' must be very small, and as these quantities are invariable, the expressions for q and r will always remain indefinitely small. ^ If n is imaginary, sin. («i+y), cos. {nt-{-y) are imaginarjs and as cos. (w«+y)+«/:iir sln.{K«+y)=C +(«^+v)-'^— 1 and cos. (n«+y)-sln. (nf+y)=c~("^ + '>')• "^—^ — we obtain by adding and subtracting I,, \ .("*+y)V^ — («<4-y).\/Zr cos. {ni-\-y) = c 4- c ^ '^ ' i sin. {nt -1-y) = c 2 («<+y). v/ZIl — («<+7)V3r if n is imaginary the preceding exponential expressions will become — nl^y s/ 1 «* — yVIZx — "'-fy\/ZI7 nt — yV~\ c Arc c _c '2 2v'zr in these exponential expressions, the part which is not affected with the radical sign, is PART I.— BOOK I. 213 axes of which the moments of inertia are the greatest, and the least j but not so about the other principal axis.* proportional to the time, and therefore the values of q and r, will increase indefinitely vvitli the lime, .•. though they may have been indefinitely small at the commencement of the motion, still as there is no limit to the increase of the exponential expressions, they will at length exceed any assigned magnitude. * It might be shewn di;-ectly by means of the equations Crp^+A^'-^-B-r-^k"; /iBC-.p^-\-A^BCq^'\-AB^Cr-=fP, that there is a limit to the increase of q and r when C is the greatest or least of the three quantities A,B,C, for if we multiply the first equation by AB, and then subduct it from the second we obtain A-.B(C — A)q'-i-AB~.{C — B).r^= Hi — AB.k^, if at any instant the quantities q, r, are very small If^ — A/fi which is constant will be very small, consequently in all the changes wliich r and q undergo they are sub- jected to the same condition, and this condition requires that r and q siiould be always very small when C — A and C — B are of the same sign, because then both the terms of the first member of the preceding differential equation will be either positive or negative, and the expressions m—AB.k^ m—ABlfi A-.B.(C—A) ' AB\C—B) ' are the limits to which the respective values of q and r can never attain. If C — B and. C — A are of different signs, then the terms of the first member of the equation will be of different signs, audit is only the difference of the quantities AiB(^C — A).q^-\-ABi, (C—By^, that is indefinitely small /. since tliis difference depends on the relative values of these quantities, q and r may be very great, though the preceding residual is' a quantity indefinitely small. Pliilosophers have distinguished the equilibrium of stability into two species absolute and relative, in the first case the stability obtains whatever may be the oscillations of the system, ;n the second case it is necessary that the oscillations should be of a certain description, in order to insure the stability of the equilibrium. If a body revolving about afixed axis passes through several positions of equihbrium, these will be alternately stable and instable. For if a system deviates from a position of stable equilibrium, from the nature of this equUibriura it tends to revert, but according as the system deviates more and more from its first position, this tendency will diminish, and at length it will tend to deviate from the original position, but previous to tliis change of tendency there must have been a position in which the systenc neither tended to revert, or to deviate from its original position, consequently this is a position of equilibrium, but this equilibrium is evidently one of instability, for previous to the arrival of the system at this position it tended to revert to its primary position, and when it passed this position, it tends to deviate from the primary and consequently from this second position of 214 CELESTIAL MECHANICS, Now, in order to determine the position of the principal axes in space, we shall suppose the third principal axis to coincide very nearly with the plane of x' and of y', so that 9 will be a very small quantity of which we may neglect the square. By No. 26, we shall have d(p—'d^ =pdt * and by integrating we obtain ^ = (p — -pt — i E being a constant arbitrary quantity. If we afterwards make sin. 0. sin. ?' = s j sin. 6. cos. <p = u ; from the values of q and of r which have been given in No. 26, we shall obtain, by the elemination of d-^ ds du , , equilibrium, this tendency of the body to deviate from'the second position of equilibrium gra- dually diminishes, and at length vanishes, afterward the system tends to revert to the second position of equilibrium, and where the tendency to deviate from the second position of equi- brium vanishes, is also a position of equilibrium, which is evidently an equilibrium of stability, for previous to the arrival of the system at this position it tends towards it, inasmuch as it tends to deviate from the second position, and after passing this third position of equilibrium it tends to revert to the second, and consequently to the third position of equilibrium, thus it appears that when a system has returned to its primary position, it has passed through an even number of positions of equihbrium, alternately stable and instable. 6* 6* * dji—d^'- cos. e=:p.dt, but cos. 6^1 — + _-&c^when 6 is veiy small, unity .•. d(p — d-^^jidt. f d4" sin. e. sin. i? — d6. cos. ^=.q.dt ; d-^. sin. i. cos. (p-^-di. sin. ^. = r.dt, substituting in place of d'^ its value dip — pdt, we shall have PART L— BOOK I. 215 and by integrating ,$ = e. sm.(p? + a) ^-=; — .(sin. nt-r y) ; tt=e.cos.(;)^+A) — ^s; — . COS. (nf+y) i* dip. sin. «sin. ip— p. sin. «. sin. ip. dt—d6. cos. <f>-:r.q.dt ; t?!?. sin. «. cos. ip— p. sin. fl cos.ip. dt ■\-d6. sin. ip = r.rff, ; substituting 6 in place of sin. 6, to which it is very nearly equal since the higher powers of * may be neglected, we obtain — d<p. 6. sin. (p-i-d6, cos. ip-\-p. sin, 6. sin. <p. rfi= — ^.t/i, i, e, d. (cos. <p. sin. () -^-p. sin. «. sin. <p. rffis — jrf^, and by substituting for sin. 6. sin. <p. sin. 5. cos. <p, their values which have been given in the text, we obtain du , in like manner the second diiferential equation becomes, d(p. 6. cos. (p+d6. sin. (p — p,dt sin. i. cos. (pzzr.dt, i, e, d. (sin, i, sin, ?i) — p.dtsin.e. COS. 9=:r.dt, and by substitution, — -^u = r. * The integrals assigned in the text are the complete values of i and a for ^= Z.p. cos. {pt4.^)-^-^A.. {C-A). {C-B). cos.(nf+v),: this expression is equal to pu-\-r, for substituting in place of u and r, we shall have ^.p. cos.(pt + A) ' P cos. (n<+y)+M'. cos. (n<+y)= (bj^ substituting for M' iteTalue,) G.p. cos. (p(+A) M /'a ~ C" •'^ 5" • ( C-^)(C— B). cos.{«<+v), 216 CELESTIAL MECHANICS, S and X being two new arbitrai-y quantities : therefore the problem is completely resolved, since the values of s and of u give the angles © and <p in functions of the time, and i)/ is determined in a function of (? and /. If e vanishes, the plane of x' and of ?/' becomes the invariable plane, to which we have referred in the preceding number, the angles 6, (? and ^. * .•• since the integrals given in the text satisfy the differential equations ds (hi and since there are two constant quantities introduced, these values of ti and s are their complete integrals. A.q . B.r * \\ hen € vanishes i = sm. i. sm. <p = — tt ~ > ii=sm. 6. cos. ip = j^ — , i, e, Cp C.p q . r' Sin. 6. sin. ® = r, sm. 6. cos. ?> = j, V P and those are values of the cosines of the angles which the principal axes of id' and if" make \vith the axis of the invariable plane, see notes to page 198. In this case s -^M , , . AM , ^ , , — =tan. <p = -g^,. tan. {nt+y) .-. <p= -^Jj' -O't + v)' as f is equal to the angle formed by the intersection of the invariable plane, and of the plane of x", y", with the axis of x", if we know this angle at the commencement of the motion, or at any given epoch, we shall have the value of y ; we might in like manner find M, for , 2 •'>,/• 2 L . ^ ^-^J" , -O-M* / A.(C—A) \ . ,^ „«+s^-s.n. -«. (sm. ^?+cos. ^):^~^^+ -^j- {'b:^c-B) ) = ''"• '' by substituting for (sin. -?-|-cos. -<f) unity, and for M'^ its value. AM 4- = -^j^r [ntJfy)—pt — e . •. as we have already determined the values of M, M', and y, we can determine the value of t, when the value of -^ is given at the commencement of the motion ; from the preceding value of •4' 't appears that tliis angle increases proportionably to the time, .•. the intersection of the mvariable plane and the plane of x" y" revolves about the axis of the invariable plane with an uniform angular velocity. PART L— BOOK I. 217 31. If the solid is free ; the analysis of the preceding numbers will determine its motion about its centre of gravity ; if the solid is con- strained to move about a fixed point, it will make known the motion of rotation about this point. It now remains for us to consider the motion of a solid constrained to revolve about a fixed axis. Let us suppose this axis to be that of x, which we will make horizontal : in this case, the last of the equations ( B) of No. 25, will be sufficient to determine the motion. Moreover let us conceive that the axis of y is horizontal, and thus that the axis of £ is vertical, and di- rected towards the centre of the earth, lastly let the plane which passes through the axis of y and of z,* pass through the centre of gravity of the body, and let us conceive an axis always passing through this centre and through the origin of the coordinates. If 6 re- presents the angle which this new axis constitutes with the axis of s ; and y^ and 2;", the coordinates referred to this new axis, we shall have y'=y". COS. e-j-s'/. sin. 9 ; s'=s". cos. 6—3/". sin. G ; from which we may obtain S. i 1^1^ l.drn = -^l. SJm.{y'- + .-). t {. at y at FF * Since the plane passing through the axis of 2', and of y', of which the former is ver- tical, and the latter horizontal, passes constantly through the centre of gravity, this centre must move in a vertical plane. t As the coordinates x", ?/' , z", do not varj- with the time, beipg always the same for the same molecule, in taking the differentials of y, s*, and their respective values, with respect to the time they become dy=di,{:i'. cos. «— y. sin.«) ; M =—dk (/'. sin. 6-\-y''. cos. i) .-. yrf*'— s'<iy =(y'. cos.«4-2". sin. 6)1— d«. (a".sin, i-\-y",cos. i) )—(*". cos. i—y'.sin. i). (di.{z!'. cos- «— y. sin. l) ) 218 CELESTIAL MECHANICS, S.dni.(i/'^-\-s{'^) is the moment of inertia of the body with respect to the axis of / : * Let this moment be equal to C. The last of the equations {B) of No. 25, will give ' dt* ~ dt Let us suppose that the body is only solicited by the force of gravity % the values of P and of Q of No. 25, will vanish, and R will be constant, which gives dW = S.Ry' .dmzzR. cos. ^.S.y". dm + R. sin.fi. S.z".dm. uc The axis of z" passing through the centre of gravity of the body, we have S.i/'.dmziO ; moreover, if we name /« the distance of the centre of gravity of the body, from the axis of x', we shall have S.z".dm = mh, m being the entire mass of the body ; therefore we shall have = —d6. {y". COS. «-f z", sin. (i)« — rf^(3". COS. «— y". sin. «)' = — rf«.(y'-+z"=) .•. multiplying by dm, and extending the expression to all the molecules we obtain, dt at and since C is constant, we shall have d^S dN" —C. dt *y*^-2" =y'-* cos.*«4-z".* sin.«+2y'z".sin. 0. cos. «+y'.* sin. M+s".' cos. '« — ZyV. sin. e. COS. « =y'^+s"« .-, S(y ?+/').(/»», the moment of inertia of the body relative to the axis of 3f=S{z!'^+j/'^).dm=C. and consequently PART I.—BOOK L 219 dN" =■ mh. R. sin. 6 * dt dH__ — m.h.R. sin. 9 dF~ C Let us now consider a second body, all whose parts are concentrated in one point, of which the distance from the axis of a/, is equal to / j we shall have for this body, C^ ??«'/* ,»i' expressing its mass; moreover h will be equal to / ; and therefore rfa9 —R . , . = — — . sm. 0 t ff2 •A?" is always equal to S.J[R7/'—Q:/). dt.dm .'. Qvankhing we shall have dN" dt "^ and by substituting for y' we obtain the expression given in the text. In fact, since the axis of s" passes through the centre of gravity, we have sy .rfw=0, and S.^' .dm=mh. See No. 15, page 91, it also appears from note to same number, page 88, that when a body is constrained to move about an axis, one of the equations (B) of No. 25, is sufficient to determine the motion of the body ; .•. by substituting mh. sin. L for sin. 6. S^^'.dm we shall have - — = mh R. sin. 6, dt f For any body m' of which all the molecules are concentrated into a point at the distanco equal to I from the axb of sf we have dH ml R . R . . for in this case the centre of gravity, is in this point, and the moment of its inertia, is equal to m'JF; if this body has the same motionof oscillation with the body we have first considered, the d^6 values of -p must be the same, i, e, mh.R. sin. 6. R . , , C 220 CELESTIAL MECHANICS, Consequently these two bodies will have the same motion of oscillationj if their initial angular velocities, when their centres of gravity, exist in C the vertical, are the same, and if we have also I = — - — * The second mh body which we have considered is the simple pendulum, the oscillations of which are determined in No. 1 1, and by means of this formula we are always enabled to assign the length / of the simple pendulum of which the oscillations are isochronous, with those of the solid which we have considered in this number, and which constitutes the compound pendulum. It is thus, that the length of the simple pendulum, which vibrates in a second, is determined by observations made on compound pendulums.t , ., „ , . d'^e R. sin. « , ^ , , . Multiplying both sides of the equation y-^= j by 2dS, and integrating we obtain dd^ 2R — = J-.COS.I+C, the constant quantity C, depends on the angular velocity, and on the value of 6, at the commencement of the motion. C ♦From the expression 1= — r-, it appears that when the axis of rotation passes through the centre of gravity, I is infinite, and consequently the time of oscillation is infinite in this case, in fact the action of gravity being destroyed, the primitive impulse will communicate a rotatory motion which will be perpetuated for ever, if the resistance of the air be removed. -)- The point which is distant from the axis of rotation by a quantity equal to I is termed the centre of oscillation of the body, and if the axis of rotation passed through this point, the centre of oscillation with respect to this new axis, will be in the former axis of rotation, for the moment of inertia with respect to the centre of gravity being equal to C — mh% the mement with respect to the new axis will be C-^-m P — 2mlh. See note, page 182, ••. the value of I for the new axis = — — "! ,"' — but C= mlk .: the value of I for the new ml — mn inp — mlh , axis = —5 ;- = I. ml — mn C'ss A sin. *i. sin. »(p+B. sin. ««. cos. *ip+ C. cos.*tf+«^^ see page 180, where J,B,C, PART I.—BOOK I. 221 are the moments of inertia, relative to the principal axis, passing through the centre of gravity, we shall have , nik^.{.A.sm.sS. sin. ^/p-^-B. sin, sj. cos. -<P-{- C. cos. 25 mh .'. I will be a minimum when the quantity represented by Cin the text is the least of the three principal moments of inertia, for in that case the other two moments vanish, let A be the least of the three moments then we shall have , mh'-i-A . . „ . „ , , . . . l^ / , lor sm. 6, cos. ip=0, cos. 6—0, .'. when / is a nummum tnh 2mVfi—7nVi^—mA „ „ , a /^ =0 .•./«= A/ — dl — „,„ .dh .-. I and consequently the time of oscillation wiU be a minimum when the axis of rotation is that principal axis, relatively to which, the moment of inertia is a minimum, and at a distance from the centre of gravity by a quantity equal to 'w — . The product of Ik. is constant C and = to — , this fraction is equal to the square of the distance of the centre of gyration m from the axis of rotation, therefore this distance is a mean proportional, between the distances of the centres, of gravity and oscillation, from the axis of rotation, and it readily appears from what precedes, that when the time of vibration is a minimum, the distance of the centre of gyration from the axis of rotation is equal to the distance of the centre of gravity from this axis, and the distance of the centre of oscillation from the same axis =.2*/— • ^ this case, the centre of gyration, is termed the principal centre of gyration. 222 CELESTIAL MECHANICS, CHAPTER VIII. Of the motion of fluids. 32. We may make the laws of the motion of fluids, depend on those of their equilibrium ; in the same manner, as in the fifth chapter we have deduced the laws of the motion of a system of bodies, from those of the equilibrium of the system. For this purpose, let us resume the general equation of the equilibrium of fluids, which has been given in No. 17» $p=f[P.Sa: -t QJ7/+RJz] j in which, the characteristic $ refers only to the coordinates «f the mole- cule X, 7/, z, being independent of the time. When the fluid is in motion, the forces which would retain the molecules in equilibrio are by No. 18, (^dt being supposed constant) ; therefore it is necessary to substitute in the preceding equation of equilibrium, these forces in place of P, Q, B. If we snipi^ose that PSx-\-Q.Sr/ + Biz is an exact variation, represented by iV, we shall have PART I.— BOOK I. 223 '^-^-(^)■^'KS^)+-c^)^•^^^ this equation is equivalent to three distinct equations ; because the variations Ss, Sy, Sz, being independent, we are permitted to make their coefficients, separately equal to nothing. The coordinates x, y, z, are functions of the primitive coordinates, and of the time ? j t let « ^ c be the primitive coordinates, we shall have we are permitted to consider PS^r-j-QSy+ii?*, an exact variation where the forces which solicit the molecules, aie those of attraction directed towards fixed or moveable points, or such as arise from the mutual attraction of the fluid molecules. We have seen in No_ 17, that this is the condition which must be satisfied, when the molecules of the fluid, are in equilibrio by the action of the same forces. f The position of a molecule at any instant, is known when we know the coordinates a, b, c, which determine its position at the commencement of the motion, or at any de- termined epoch, .-. X, y, z, are respectively functions of a, b, c, and t, consequently «e have x-=zf{fi, b, c, t),t/=:F.{a, b, c, t,) ; zzi<p.(,a, b, e, t). and as the differences indicated by the characteristic S refer solely to the variations of the coordinates n,b,c, being independent of the time, the expressions for dx, lij, ^z, should be such as are given in the text, .'. if it was pro- posed to compare the respective positions of two molecules at any given moment, the tiane should be considered as constant, and the expressions for ^x di/ iz should be those which are given in page 22i, on the other hand, if we consider the motion of the same molecule for the time di, the values of dx, dy, dz, deduced from the preceding expressions for x, y, z, must be taken on the hypothesis that t only varies and . •. when t=0, x=.a, ij=b, z-=c. If the form of the preceding functions was given, by eliminating the time from the equations which determine values of X, y, z, the two equations which result will be the equations of the curve described by the molecule, however as a i c are different for each molecule, the nature of this curve and its position will be different for each molecule, see Note page 31. Q24 CELESTIAL MECHANICS, Lda ) cao) (ac > By substituting these values in the equation (2^), we may put the coeffi- cients of Sa, Sb, Sc, separately equal to nothing ; which will give three equations of partial differences between the three coordinates of the molecule x,y, z, its primitive coordinates a,b,c, and the time t. It remains to satisfy the condition of the continuity of the fluid.* For this purpose, let us consider at the commencement of the motion, a rectan- gular fluid parallelepiped, of which the three dimensions are da,db,dc. If we denote its primitive density by (p), its mass will be equal to {p).da.db.dc. Let this parallelepiped be represented by {A), it is easy to see, that after the time t,i it will be changed into an oblique angled parallelepiped ; for all the molecules which in the primitive situation existed on any face of the * In order to determine the condition of a fluid mass at each instant, we must know the direction of the motion of a molecule, its velocity, the pressure p, and the density g, but if we know the three partial velocities parallel to the coordinates, we shall have the entire ve- locity, and also the direction, for the partial velocities divided by the entire velocity, are pro- portional to the cosines of the angles which the coordinates make with the direction, see Note page 26, and page 227. Three of the equations which are required for the determination of those sought quan- tities, are furnished by the equation (F) ; another equation from the continuity of the fluid, for though each indefinitely small portion of the fluid changes its form, and if it is com- pressible, its volume during the motion, still the mass must be constant, consequently the pro- duct of the volume into the density must be the same as at the commencement,. • . by equating those two values of the mass, we obtain the equation relative to the continuity of the fluid. f After the time t, the coordinates of the summit of the parallelogram, whicji were a, b, c, at the commencement of the motion, will be j:, y, z, ory(a 6c<), ^{a i c <), ip (a 6c/), the coordinates of that point of which the initial coordinates were a, b, c-^-dc, will be y PART I.— BOOK I. 225 parallelepiped {A) will still be in the same plane, at least if we neglect quantities indefinitely small of the second order ; all the molecules si- tuated on the parallel edges of (A) will be found on small right lines, equal and parallel to each other. Denoting this new parallelepiped by (B), and conceiving that through the extremities of the slice constituted, of those molecules which in the parallelepiped (A) compose the side dc, we draw two planes parallel to the plane of x and i/. Then producing the edges of the second parallelepiped to meet these two planes, we shall have a new parallelepiped (C) contained between G G f(a, b, c+dc, t), F(a, b, c+dc, t), (p (a,b, c-(-dc, t)z= respectively to . the difference between these coordinates and x, y, s, are and the square root of the sum of the squares of these three quantitities, is the value of the side of the parallelepiped which answers to the side dc of the primitive parallelepiped; extracting the square root, and neglecting the third, and higher powers of dc, this side becomes equal to dz drz , Jc-'^^+a— •^'^' in like manner it may be shewn that the quantities which in the original parallelepiped are equal to da, di, become the opposite sides of the figure are equal to these ; for the value of x, y, z, which corresponds to the primitive coordinates a-^-da, b, c, are/(a+da, J + c t)F{a-\-Aa + b,ct,) ?i(a+da, b c f)= x+J.da+^.da^^+|^.da+g-,.da^ .-j- ^.da+^.da^ da ^%da^ ^ ' da ^2.da^ ' ^ da 2,dai 226 CELESTIAL MECHANICS, those planes, and equal to (B) ; for it is manifest that what one of these planes takes from the parallelepiped (B), is added by the other plane. The two bases of the parallelepiped (C) will be parallel to the plane j', t/, : its altitude contained between its bases will be equal to the difference of ~, taken on the hypothesis that c * only varies ; consequently this altitude will be equal to | — V dc. the values of x, y, z, which answers to the primitive coordinates a-\-da, b, c-j-dc, will be y"(«-j-da, b, c+dc, t) F(a.+da, b, c+dc, <) ifi(a-j-da, b, c+dc i)r: da 2. da- ' dc ^ <2..d(? ' ''da ^ ^da- 'dc idc^ dz , d-z ^ g , dz , d*z . , '~'^do^'+2d^-^^'^d-a-^^+2i;^- ^^' .'. the difference of the coordinates of these points = -7-.dc+ — -. dc^ -f. dc+ -^_. Ac-, -. d<;+— =-:-. dc?, dc IMc- .dc ^ Idc^ dc ^2c?c* and as these differences are equal to the corresponding differences of the opposite side of the figure, it follows that these sides must be equal, being equal to the square root of the sum of the squares of these differences, in like manner it may be proved, that the other sides are respectively equal to those to whichjthey are opposed ; and the parallelism of theee sides is a necessary consequence of their equality, fiom which we infer that the figui-e wliich the molecules assume is a parallelepiped. The equation of the line connecting the points whose respective coordinates are f{a,b,c,t), F{abcl), (p(a bct),f(a-\-da,b ct), F(_a+da, b, c t,), (p)a-^-da,b ct), will be that of a right line, if we neglect the indefinitely small quantities of the second order, and the same is true for all lines parallel to this line, of the sum of which the face may be conceived to made up, .•. this face may be considered as a plane. * The difference between the values of z corresponding to the expressions .=,ia,b,c,t),-J=^,.^abc+dct)^'^. dc+ ^£}.g=5|}.dc PART I— BOOK I. 227 We shall obtain its base, by remarking that it is equal to a section of (B) made by a plane parallel to the plane of a:, y, ; let us designate this section by ({)• The value of z will be the same for all the molecules of which this base is constituted, therefore we shall have °= &3-^- tiM'^rX- ''■ Let Sp, Sq, be two contiguous sides of the section (e), of which the first is made up of molecules which existed on the face Ab. dc. of the paral- lelepiped {A), and of which the second is composed of molecules which existed on the face da. dc. If we conceive two lines to be drawn through the extremities of the side Sp, parallel to the axis of .r, by pro- ducing them to meet that side of the parallelogram (f), which is parallel to Sp, they will intercept a new parallelogram (x) equal to (t), of which the base will be parallel to the axis of x. The side Sp being composed of molecules which existed on the face d6. dc, and relatively to which the value of ~ is constant ; it is easy to perceive that the altitude of the parallelogram (x) is the difference of y, on the supposition that a, z, and t are constant, consequently we have ((Iz ) db+ 5 7— f . dc; (dc i G G 2 by neglecting quantities indefinitely small of the second order. For all the molecules situated on the edge, which corresponds to dc in the original parallelepiped, projected on the axis of z, the values a and 6 remain the same, nor do any molecules which occur in the face daM enter in the constitution of this perpendicular, therefore it is equal to dz on the hypothesis that c only varies. * If we conceive the molecules of the face db.dc relatively to which dz is constant, to be projected on the axis of y, it is evident that the projected Lne is equal to the difference 228 CELESTIAL MECHANICS, from which may be obtained a this is the expression for the altitude of the parallelogram (x). Its base is equal to a section of this parallelogram by a plane parallel to the axis of X ; this section is composed of those molecules of the parallelepiped [A), with respect to which z andj/ are constant ; its length will be equal to the differential of x taken on the hypothesis that z, y, and t are con- stant, which gives the three following equations "'-XTi^'^YiV^^tW- Ida C Idb^ idcy -{i}-ii}-^^+{^l-- of y, on the hypothesis that a is constant, for this projection is the same for every series of molecules, which exist on the face which corresponds to the primitive face di.dc, and rela- tively to which z is the same. We obtain the expression which is given in the text for d^ by eleniinating dc between the two preceding equations. * Since the parallelogram (a) exists in the plane parallel to the axes of «, y, the value of z will be constant for this parallelogram, and since the base of (a) is a line parallel to the axis of a the value of y ■ndll be the same for all molecules situated in this base, but since in this base molecules occur which belong to the faces da.dh, da.dc, db.dc, a,b,c, will vary for these molecules. PART L— BOOK I. 229 In order to abridge, let us make ~ Ida ^- Idb^'^dc \ lda\'ldc\'ldb^ idxl (di/l (dz^l , ^ Idb^'ldc^'lda^ • Multiplying the second equation by j j- f > and the third by -! ;r f . and then subtracting we shall eliminate dc ■■■ |(i-)(i)-(|)L^)}-«+ |(l)(S)-(|)(|)}-a*.=o " ' , , ^ , , .da in like manner we can obtain {{7:}-{l}-{l}-{^i}'-+{{l}-{|}-{|}-{f}}-=o .•.dc= \di>s'\d^s~ \Ey\da s ^^ dx \ da' ^dz\ Sdj\_UyX idz^y-Xdb\ ^ XdcS'XdbS \dcS'\dbs 230 CELESTIAL MECHANICS, ldb\'lda\'ldc\+ ldc\-lda\'ldb\ Ucylm'idcS we shall have Q.da dxzz IdbS Idc) Idcf'l dz_ db this is the value of the base of the parallelogram (a) j therefore the isur- face of this parallelogram will be equal to ^Aa.Ab \dc) This quantity also expresses the surface of the parallelogram {i), if we idz' multiply it by4(—/dc we shall have ^AaAbAc for the volume of the \db J • ^f/a5 \dly If/a 5 ( dx~\ Sdz-x Sd_y \_ Sdy\ frfc-i ' \ del \dc\'\db j \dci '\dh] , da= {l^}{S}{i}-{^:}-{l}.{i}+{|}.{f}.^,t] • XdcS'XdbS Xuci'Xdh) Q. da \dcj\db\ idcfidbi = the base of the parallelogram (a), this expression being multiplied into the value of di/ gives the^area of (a), and this area being multiplied by the altitude gives the volume of (Q PART I.— BOOK I. 2Si parallelepipeds (C), and (5). Let p represent the density of the paralle- piped (A), after the time /; we shall have its mass equal to p Q.da.db.dc j and by equating this to its primitive mass {f).da.db.dc we shall have pe = (p); (G) for the equation relative to the continuity of the fluid. 33. The equations (i^) and (G) may be made to assume another form, which is in certain circumstances of more convenient application. Let u, V, and V be the velocities of a molecule of the fluid, parallel to the axes of X, of y, and of z j we shall have {ll = -{'f|=-l.T} = ^- By differentiating these equations, u, v, , V being considered as functions of the coordinates x, ?/, z, of the molecule, and of the time t, we shall have c?'.r>_ (du\ <f^in . (din ,j cdu-i j d\t *«» w> V, are respectively unknown functions of x,y, z, and /, they depend on the coordinates X, y, z, because for a given value of t, the velocity is different in different molecules, they depend on t, because for the same values of x,y, z, the velocity varies every instant, •••-=m-'M<^}'^+{|}''-+ {£}•*■ and since dx=udt, dy = v.dt dz='\dt, substituting and dividing by dt, we obtain , . dx da d^x but u = — .: — = — . dt dt dt^ n .1, 1 J-*'" dv dW . . , trom the values of^,^ ,— , given m the text, it appears how the increment of each of the three velocities depends on the two other velocities. F we were able to determine the 232 CELESTIAL MECHANICS, consequently the equation (F) of the preceding number will become. In order to have the equation relative to the continuity of the fluid ; let us conceive that in the value of S, of the preceding number, a, b, c, were equal to t, i/, z, and that j;, y, z, were equal to x + udt, y+vdt, z+V.dt, which is equivalent to assuming the primitive coordinates a, b, c, indefinitely near to cc, i/, z, j we shall have value of !4 in a function of x, y, z,t, we could by means of the equations ~jj'— "' ^ = '"' dz_ It' position of this molecule, and also what function of x i/zt, uv\ are, for substituting in the dx dy dz _ ^V determine the position of a molecule at any instant, provided we know the initial dt iition I equations — = ti, — = v, -j =V the values of t« v, V, in functions oixyxt, and integrating, ^ dt dt dt we would obtain the values of «, y, x, respectively in a function of i, the constant arbitrary quantities which are introduced are the values of «, y, z, at the commencement of the motion which by hypothesis are given, consequently the values o( x y z will be completely deter- mined for any instant. Eliminating t between values of x, y, z, to which we have arrived, we would obtain the two equations of the curve described by the molecule, but since the initial position of each molecule is different, the form of this curve will also be difterent, as will be in like manner, the position. PART I.— BOOK I. 233 :dV-) HH • The fii-st coordinates being assumed indefinitely near to x,y, ~, we shall have da — ds, and the quantity which corresponds to rfa:=to (/«+*«.*, in like manner we shall have dx,+du.dt dx+diuft di/-\-dtrdt di/+dv.dt dz+dV.dt dz+dV.dt "~d^ ' dy ' dz ' dz ' dy ' dx respectively indefinitely small, because when t—O these quantities vanish, /. the product of any two of these quantities may be neglected, making these substitutions the expression for C becomes equal to (dx + duJl\ fdi/ + dv.dt\ (dz+dYdt} \ dx ]•{' dy / ■ t "~d2 ) _ (dx+du.dt\ fd^-hdv.dt\ (dz + dy^t\ \ dx M" rf-' ri dT i , cdxj-dtudt-i (dy+dv.dty cdz+dV^l "^ t dy r\ dz ]l dx i f dx-\-du.dt ■) f dy+dv.dt ■> r dz+dV.dt 1 1 dy rl dx i'l dz y (dx-\-dn.dl\ ( di/+dv.dt \ cdzJ-dVJl) "^1 dz r\ dx i'\ dy i i dx-\-du.dl-i ^dy+dv.dt\ f dz-^dY.dt \ I dz \ \ dy ]\ dx \ the first term of this expression = by neglecting quantities indefinitely small \dx dy dzl the other terms of this expression vanish. It appears from what precedes that €j is a con- stant quantity independent of the time, when the fluid is incompressible S=l. 234 CELESTIAL MECHANICS, the equation (G) becomes, If we consider p as a function of x, y, z, and t, we shall have therefore the preceding equation will become * The density {, the pressure^, may be shewn to be functions ol xy z, t, by reasoning, analogous to that, by wliich u, v, V, were proved to be functions of these quantities ; is the increment of g on the supposition that t is constant, "'• {1} -«•*■ {1} -* {|} -^- {S} is the variation of § on the hypothesis that x, y, z, t, vary .*. their difference is the differential of the equation (fi) taken with respect to the time ; PART I.— BOOK I. 3SS this is the equation relative to the continuity of the fluid, and it is easy to perceive that it is the differential of the equation (G) of the pre- ceding number, taken with respect to the time t. The equation (H) is susceptible of integration in a very extensive case that is, when uSa: + vJj/ + YJz is an exact variation of *, t/, z, p being any function whatever of the pressure p. Therefore if we re- H H 2 when the fluid is incompressible, we have for in this case both the magnitude, and density are constant, .•.</£ and d^ are re- spectively equal to nothing, these two equations combmed with the three, which may be derived from the equations {H), or (f ), are sufficient to determine p, {, and the three partial velocities, u, v, V, in functions of x, y, z, t,. When the differential coefficients -i, - ,— , — , vanish of themselves, g must be a constant quantity, and the incompressible dt dx dy dz fluid will be also homogenous, .-. in this case the number of unknown quantities is reduced to four, which is also the number of differential equations. When the fluid is elastic the number of unknown quantities will be ultimately reducible to four, for when the temperature is given /)=yt. g, .-. the equation {K) and the tliree equations (i/Jare sufficient to determine the unknown quantities, in this case ^1—L h.— l- S. log 5. k will not be constant when the temperature varies, but if the law of its variation is known, since for each different instant, and point of space the temperature is a given fonction of x,y,z,t,}c will be so likewise, so that even in this case the equations (A') and H are sufficient to determine ^,u,v,\. It appears from what precedes, that we have always as many equations of partial differences as sought quantities, however the general integration of these equations has baffled the ingenuity of Pliilosophers and even granting that it is possible to effect this integration, still the determination of the arbitrary functions introduced by these integra- tion, is extremely difficult, these functions depend partly, on the primitive state of the fluid, and partly on the equation of the exterior surface. 236 CELESTIAL MECHANICS, •present this variation by $<?, the equation (H) will give from which may be obtained by integrating with respect to i, * If we take the differential of the equation u2*:+v2i/-\-Y .^z with respect to i, x, y, z, we shall obtain '>*+ £-^-*+ E-^-'-- 1-^'=£- -"'^ t-""'^Z -'•■•" «« dz -^ ' dz dz dz c/z ^ ' dx- now substituting udl, vdt, Ydt, in place of dx, dy, dz, and remarking that, j^— j-, v=z —, &c. and also that 3.-? = _1^ we shall have dt dt := the sum of the last members of the preceding equations, but these by concinnating, and dividing by dt are evidently equal to the second member of the equation ( H). Since the integration is only made relative to the characteristic S, it is evident that the time is not in- volved in this expression. When the fluid is homogenous —&c.=0.*. the equation of con- tinuity is reduced to the second term, by means of this equation, and the equations uzz. -j-, PART I.— BOOK I. 237 It is necessary to add to this integral, a constant quantity, which is a function of t ; but we may suppose that this function is contained in the function (p. This last function gives the velocity of the molecules of the fluid parallel to the axes of .r, of y, and of z ; for we have The equation (AT) relative to the continuity of the fluid, becomes consequently, we shall have in the case of homogenous fluids, It may observed, that if the function u^s + vS^ + YJz is an exact va- riationof a:, t/, z, at any one instant, it will always remain so. In fact, let us suppose that at any instant whatever, it is equal to Sep, in the sub- sequent instant it will be equal to '^^^'■p\-^^i\'>'^m>^] . * D=— ?, V= — , and the value for y"-^, r: in this case — , we can determine (p and p and dy dz 5 5 consequently u, w, V, in functions oi xy z. • From the value of V — f. — it appears that the pressure of a molecule, of which the e density is constant, diminishes when the velocity which is equal to 238 CELESTIAL MECHANICS, therefore it will be an exact variation at this new instant, if r}''-i:f}-^+{.?}- :clu ^m'^Yi\'+\t] is increased. substituting this value of S. J ^ < b the expression for JV— £ we obtain and since each of the terms, of the second member of this equation, are exact variations of m, y, z, the first member will also be an exact variation, we suppose g to be a function of p. is the differential of S0, on the supposition that the time only varies. Consequently, we are not obliged to determine ip in j:,y, z, in order to know whether it is an exact differential or not. .•. It appears tliat if ii^x.\-v1y-{-V .^z be an exact variation, at the subsequent instant tts increment will bean exact variation, .-. S?i + this increment will be an exact variation. As in general we know the condition of the fluid at the commencement of the motion, if at this moment t(Sx+ uJj/^-V.Si is an exact variation, it will be an exact variation when *^ ± df, t~ ± 2dt, &c. and in general whatever may the value of t. ?<?x+ v.Jy+ V.Ss will be an exact variation, if when t~0, the fluid either has no velocity or a consUmt one, for in first case u=0,v—0,V=0 when t vanishes, .-. «?jr+t)Sy+VSz will be integrable for this moment, the second case will obtain when the motion is produced by an impulse on the surface of the fluid, such as that which arises from the action of a piston. For the velocities u, V, V which are communicated to each of the molecules, must be such, that if they are PARTI.— BOOK I. 239 is an exact variation at the first moment, but the equation (H) gives at this moment consequently the first member of this equation is an exact variation of X, y, z, ; therefore if the function uSx-\-v.Sy+W.dz be an exact variation at any one instant, it will be one in the next, therefore it will be an exact variation at all times. When the motions are very small; the squares and products of «, v, V, may be neglected ; and the equation {H) will then become therefore in this C9.se uSx + vSy + YJz, is an exact variation, provided that, as we have supposed, ^ is a function of p; therefore if we designate destroyed by impressing on each molecule, equal velocities in an opposite direction the entire fluid would quiesce ; .'. in consequence of the primitive impulsion, and the velocities u, v, V, applied in an opposite direction, there must be an equilibrium, .*. m ti V must be such that M3«+i'Jy-|-V.Sz may be an exact variation, see No. 17 ; it appears from what precedes, that the integrability of the equation (//), and the consequent determination of p, g, a, f, V, depends on the nature of the velocities, communicated to the molecules at the commence- ment of the motion. * In the equation {H) u, v, V, are very small quantities, and in like manner .'. their product may be rejected .*. naming this variation 3ip we have as before, '4'-='S-'-={{.t)-'^(.t)-'»+(?)-'-}- ' 240 CELESTIAL MECHANICS, this variation by i(p, we shall have and if the fluid be homogenous, the equation of continuity will become lc/a,'^S UyA ^dz*S the expression o^^m+0'^o' is the value of V — / *, when uSx+v^y-^-V .h is an exact variation, it is reduced to e its first term when u, v, V, are very small quantities. However though the form of these equations is comparatively so much simpler, than the general equations which have been given in page 232, still the determination of the lav>s of the small oscillations of the waves of the sea, is yet a desideratum in Physics. Philo- sophers have been much more successful in investigating the oscillations of the pulses of the air, and in the determination of the velocity of the propagation of sound. Tlie integration of which is the equation relative to the continuity of the fluid, when wJx+v.Sy+VS* is an exact variation, and when the fluid is homogenous, which is consequently the simplest possible form, is extremely difficult, however it has been completed effected by Antonie Parseval, PART I.— BOOK I. 241 these two equations contain the entire theory, of the very small un- dulations of homogeneous fluids.* 1 1 • If the fluid which makes small oscillations be water, by making the axis of z vertical, fl>z=g.3r,g representing the force of gravity, Pdx, Q3y are= respectively to nothing, in like manner we may cortteive it to be homogeneous and incompressible, consequently we shall have /i=£....,._fe=,.{*)=,.,^'A.,^ f = O- at the surface p vanishes, •'•-="• 1 j7 I ' consequently when the form of ? is deter- mined, we can derive the equation of the part of the fluid in which p=0, i, e, the equation of the surface of the fluid. We determine <f as was already observed by means of the equation m^i^vv^}-"' For, elastic fluids or those whose density varies, p^zt §, and if (j) the density of the fluid in a state of rest, becomes in a state of motion equal to (?)+(?)■?» 9 being a very small quantity, 5 will be equal to (5) + (^). J, the oscillations being supposed very smaU, •iV~^£zz'i. ^^jwai become 3 V—i, ^=:J. i ^ ] , the only force acting being that of gravity, and the motion being supposed parallel to the horizon, 3 V will vanish and the equation will become — -^ = ?. •! ~ j- = by substituting for { its value, ({) being sup- posed constant,— '-jr-^; •••— 1. log. q= \^.\ .the equation relative to the continuity of the fluid will become vanish, the motion being supposed to be performed in a direction parallel to the axis of s, and 2*^ CELEStlAL MECHANICS, 94. Let us consider art homogetieous fluid ifiass which i-evolves Uni- formly about the axis of x. n represeilting the angulal* velocity of rotation, at a distance from the axis equal to unity, we shall have v = —'nzy Yz=.ny; * consequently the equation [H) of the preceding number, will become P consequentiy the velocities v,V, = respectively which is a quantity indefinitely small of the second order, /. it may be neglected, consequently the preceding equation becomes « \-o>^Mm =»■ ^"•^— ■'^■'- ^--{^} con this equation is of great celebrity in the history of the integral calculus, it was first in- tegrated by D'Alembert, in an analysis of tlie pfoblem of the vibrating chord, which leads to an equation of precisely the same form. * The linear velodtyis equ&l to the angular velocity mnkiplied into tlie distance, .-.at a distance represented by unity, the linear velocity =n, and since the angular velocity at all distances from the axis is the same, at a distance=v' ~z^~+p' the linear velocity = n. V 22+^1, the direction of the motion being perpendicular to the radius in order to obtain the velocity parallel to the coordinates r,y, we should multiply n. \/z»+^» into the cbsiiies of the ailgfes Vhich z and^'make with the "tangent, but these cosifles are respectively y — z /. „;. ~„' — „' for the motion being circular, if one of the cordinates be increased, the other will be diminished .•. v=z nz, Y=ny. t The tenns torr^pwdfng to -{ t- f > j -7- 1 , \ -r- \ .in the equation (i!/) vanish, because the time does not enter into the values of u, v, V.in like manner a and its differential ooefficients vanish, and from the values of v, V, given above, it is manifest that PAItT I.— :^00,K I. 1^3 wiiich equation is possible, because its two members are exact yariations. The equation (^K) pf tjie same nu^iber will become and it is manifest that this equation will be satisfied, if the fluid mass ,be homogeneous. The equations of the motion of fluids will therefore be satisfied, and consequently, the motion is possible. The centrifugal force at the distance 4/3/* +z* from the axis of ro- tation, is equal to the square Ti'.(_^+i/^) of the velocity, divided by this distance; therefore the function n^.(i/Si/ + z.Sz)i is the product of the II 2 /— ), ( — -},are equal respectively to nothing, consequently the only terms wliich have a finite value are V. \-f-)> '"•V j~)' which are respectively equal to zz-r-n^y, — «' «> •••the equation (H) will become ^^ SF+n^(v5^+^Sz), this equation determines the pressure e when ^ is- constant, «r. when it is a function of p, * The equation (K) is resolvable into two parts as before, (I) +«-(|)+- (|)+-■(l)+^{(|)+(|)+(£^)^ the velocity being uniform, its increment resolved parallel to the axes of x, y, z, i, e .du \ ( ^'v\ /dV . {di)'^~d^)'{oiry must be severally equal to nothing, this is evident for v, V, from their values which have been given above, with respect to the velocity u, it must be produced by the part of the velocity which is parallel to x, and if it was not uniform, the fluid would not have a uniform motion of rotation about the axis of x, zdz-{-yh/ t The centrifugal force = ?j\\/ 2-+ v^ the variation of the distance = , •■• "'•(z^^ + ySy) is = to th^ centrifugal force mi^ltiplied into the element of the distance. S44 CELESTIAL MECHANICS, centrifugal force, by the element of its direction ; thus, if we compare the preceding equation of the motion of a fluid, with the general equation of the equilibrium of fluids, which has been given in No. 1 7, we may perceive that the conditions of the motion are reduced, to those of the equilibrium of the fluid mass, solicitedby the same forces, and by the centrifugal force which arises from the motion of rotation ; which is sufficiently evident from the nature of the case. If the exterior surface of the fluid mass be free, we shall have Sp—Q, at this surface, and consequently 0 = SV-^n^.{ySy-\-zSz) ; * Substituting for SFwe obtain -!—=^P.^x-\-QJy\-R.'^z-\-m.yly\-rfiz.tz,t]\e quantity added i, e, the centrifugalforcemultiplied into the element of distance, being an exact variation, itfollows that the expression for —will in this case be an exactvariation, n is some function of the distance of the molecules from the axis of rotation, as the tme is not involved in the preceding equation, it follows that the conditions of the motion of a fluid mass, about an axis, with a given velocity, are the same as the conditions of equilibrium of a fluid mass, the same forces as before soliciting the molecules, combined with the centrifugal force, arising from the uniform revolution about the axis. The molecules of the fluid, though they have a motion about an axis, are relatively at rest. * At the exterior free surface Sp=0, .*. 3 F+n -(?/Jy + 2J2)=0, .-. in order that the form of the fluid, may remain the same, during the entire motion, n must be constant. If die fluid was water contained in a vessel open at its upper surface, j is constant, and 3 V=g.2x the axis of rotation being supposed vertical, .•. Q.Sy, iJSz vanish, and P=g, consequently, we shall haveZ.^ — gx-j-n^.i " "^^ j+/«and at the free surface, wehave.T=w".^ ~^-^ j -| for the equation of this surface ; if m*. \/ z^ + if which expresses the centrifugal force varied at the 2r — 1 power of the of the distance from the axis of rotation i, e, as 2r— 1 2. r— I (2*+/);" ' =a ». (a »+i^ -), and/«^0^^+zSi) r ' \ ^r I' ^ "ir.g ) g PART I.— BOOK I. 24J from which it follows that the resultant of all the forces which actuate each molecule, must be perpendicular to this surface, moreover it must be directed towards the interior of the fluid mass. If these conditions be satisfied, an homogeneous fluid mass will be in equilibrio, whatever may be the figure of the solid, which it covers. The case which we have discussed, is one of those in which the variation uSx + vSy-{-YSz * is not exact ; for then this variation becomes .•. if r is positive, x is least, when (2*4-2^) =0, when r =1 all the molecules revolve in the same 2^-4- 2/ \ li time, and *= a * . i ^ J -i — which is the equation of the concave surface of the parabo- loid, of which the parameter= — — , the periodic time being equal to the force divided by the distance = — . .•. if the time of revolution, be called T, we shall have the parameter of a the generating curve rsto —T'Sc— = — ~ Zj/ - +A — ps .*. x being the same, the pressure is gi'eater at a greater distance from the axis of rotation. When r is negative, at the point where i*+y2 =0, x is infinite, and when= — h the surface of the fluid will be such, as would be generated by the revolution of aconical hyperbola, about its asymptote, the axis of x is in tliis case the as)Tnptote. The constant quantity h denotes the distance of the origin of the coordinates from the other asymptote, .*. both in this case and where the surface of the fluid is paraboloidal, the constant quantity depends on the quantity of water in the vessel. If the vessel was cylindrical, we could determine the area of the paraboloid, provided that we knew the area of the base of the cylinder, and also the points of greatest elevation and depression, for the paraboloid is half the circumscribing cylinder. This paraboloidal figure is that which is assumed by the molecules of the fluid, in the ex- periment which Newton adduces, in order to shew that the effects by which absolute and relative motions are distinguished from each other, are the forces of receding fi-om the axis of circular motion. See Princip. Math, page 10. • wJx-t-uJy+ V.J2 is not an exact variation in the preceding investigation, for substituting for V, and V, we obtain t,=_«;,V=ny, . •. wJx-f- v.ly+ V.S2=n.(!/Jz— z.Sy), consequently it appears, that though the circumstance of the preceding expression being an exact variation, would facilitate very much, our investigations, still it is not essentially necessary, that this should be the case, in order that the motion should be possible. :■ Since in the case of the sea, revolving round with the earth round its axis, and relatively quicscing with respect to the 246 CELESTIAL MECHANICS, ^^n{zh/—ySz] ; therefore in the theory of the flux and reflux of the sea, tve are not permitted to assume, that the variation concerned is exact ; since it is not so in the very simple case, in which the sea has no other motion, but that of rotation, which is common to it, and the earth. 35. Let us now determine the oscillations of a fluid mass which covers a spheroid revolving about the axis of t; and let us suppose that it is deranged from the position of equilibrium, by the action of very small forces. At the commencement of the motion, let r represent the distance of a molecule of the fluid, * from the centre of gravity of the' spheroid over which it is spread, and which we shall suppose immoveable ; let 6 be the angle which the radius r makes with the axis of a:, and zr the angle which the plane passing through the axis of x and the radius r, constitutes with the plane of x and of j/. Let us suppose that after the time t, the radius ?• is changed into r + a,s, that the angle fi is changed into 9 + aw, and finally, that the angle t3- is changed into 7it+-By + a.v; a.s, aw, and af, being very small quantities, of which the squares and products may be neglected, we shall have x = (r-\-cis). cos. (6 + Ml) } ^ = Cr4-«s). sin. (9+«m). cos. (n?+ in- + aw); 2;:=(r+a5). sin. (S + aw). sin. {nt-i-zr^-aV). eartb, u'ix-\-vii/-\-y.h is not an exact variation, we may conclude a.Jbrtiori, that it is not one, where the oscillations arise from the attractions of the sun and moan, which produce tlie flux and reflux of the sea. In order to ascertain whether an incompressible fluid solicited by accelerating forces, 'and also by a centrifugal force, may be at the surface of a given figure of revohciion, wA substitute in the equation 0=^V+n'^{i/di/+z.'iz) the forces parallel to x, y, z, which would result from this hypothesis, the resulting expression should be the differential equation of the given surface, if it is not, then we may be certain that the given curve does not satisfy the equilibrium of the fluid. See Book 3. Chap III. No. 1'8. * If a perpendicular is let fall from the extremity of r on the axis of a-, it will be equal to ''r; sin. (, and the projection of this perpendicular on the plane ofij,x, is equal to the coordinate y and its value will be r. sin. «. cos. -ar, and this perpendicular projected on the plane s x will be the coordinate •«, and it will be equal to r. sin. i. sin; «t. PART I BOOK I. 247 Substituting these values in the equation (^F) of No. 32, we sh^li obtain, the square of « being neglected, * * Since xu, »^, ««, are very small quantities, of which the squares and products may be neglected, the time t will of the same order as «, so that at is of the order a. *, consequently sin. tcu := »u — &c.= xu, COS. «a=l — - — — =1 .•. x = {r.-{-»s). qas. («+««) ^r. COS. 6. COS. au — r. sin. i, sin. au-^-»s. cos. ^.cos, t^u — tts. ^in. I. sin, »u = by neglecting quantities of the order <«*, r, cos. 6 — r. sin. 6, ccu+*s. cos. 0, r and t are independent of t, dx du . , , ds . d^x d^u . , . d*s .•.-— = — y .ar. sm. tf-f- -T- «. cos. J ; — — = — -r— xr. sm. t-\- rrr^. tt. cos. t, dt dt ' dt df dti ^ dt^ (d'x \ ■d*u id^s <f*K d's = — h:ra.wa. t.cos.e.- u3r. a. cos. *S.- — l-3tf. r*«. ad. 'I'-, )^.r«.sin.J.cos.«.-r-, df ^ dt* df dt' rejecting quantities involving «* &c; ^ t=(r-\-ics), sin. (*+«m). cos.{nt+tt-{-»v)=r. Bva.((-iritu). cos.(wf^«+«») -f-(M. sin.(«+«a).cos.{n<+'5r-f «ti)=r. sin. 0, cos.(nt-\-ir) — r. sin. «.sin.(«r+n<)«u -f-teu r- cos. 0. cos.('srf-n^)+«i. sin. *. cos. (■et+''0 rejecting as before quantities of the order <**, substituting »u, civ, for sin. ^^!^ sin. ««;, and observing that at is of the order «*, ,*. yz: r. sin. *. cos.yv — ntr. sin. ». sin. a-.— r. sin, ^ sin. <iir cui—nrtitv. sin. <. cos. ■a + aur. cos. 0, cos, w •-^«urn^. COS. 4. sin. '^ir -fxvf • sin. I. cos. .jr-^oi^^ sin. (• .sin. 'et ; dv • • . ■dv . dv . •^= — »r.«m.f<.sm.»!— r.sin.tf. sm.^.-a.'r^— T»r. «v. sm. *..(!flS' ■wrr-w*^. 5-.sin.«.cos.« dt dt dt dv. . . ^ *'« , .• <«r. cos, f.-coa. «. — r— «t*moos.#. ein.w— «rrrf. -r-— cos.**sui.«r 248 CELESTIAL MECHANICS, «r«.<r9 J (^)— 2w. sin. fi. cos. 6.(^) } +ar'.jTff. )sin.^9.(^— |-) +2n. sm. 9. cos. 6. \-r) + ~"\T)i > ^^) = ^. ,J.j/'ri-«5).sin. (9 + «mU +(^7— A rfs . . . . . (is 4- II.-T-. sm. 4. COS. «r— «$ s. sin. 4. sin, « —»nt. sm. ^. sm, «, -r- at at d*y ■ . ■ '^^'" o -A ^'^ I . d^" — i= — ar. sm. 6t sm. w.-rr- ■— Z«r. at. sm. t, cos. <r. -r- +«r.COS. 6. COS. «7. -; — — 2«>*n. COS. ^ sm. w.-r- + <c. sm. (>. cos. «r. -^ — 2«n. sm. t, sm. v-^j — 3y=3r. sin. «. cos. iir-|-9^, r. cos. i. cos. «t — Jw. r. sin. <. sin. -a, rejecting those quantities in the value of iy, where <e occurs, for in the product of the ex- d*ii pression for ^ into the value of 3^, these would be of the order «s*, .«. they ought to be neglected ; .3v. — -=lr.( — «r. sin, '«.sin.«. cos.<Er)-— 2nr«. sin. ««. cos. -«r, ^ dt^ ■ dt* dv dt d'^u . . du + «r. sm. (. cos. i. cos. * v. -; 2«rn. sm. t, cos. ^. sm. •a. cos. w. — — (/<» dt + «. sin. *tf cos. *«'-T-; — 2<t».(sin. *«. sin. cr.cos. ^)-'7t ) 3^(— «»•' sin. 6, COS. *. sin. w. cos. <a 2nr*a.wa. <.cos, i, cos.*c.-t- + <tr*. COS. **. COS. 'c-T- 2«nr*.cos,*^sin, «, C09.W-; — H«r. sin. <.cos. *. cos.'cr-— ^ PART I.— BOOK I. 249 At the exterior surface of the fluid we have ip=:0 ; moreover in the state of equilibrium, o= "-S.[(r + ccs), sin.(9 + ««;}* + (<Jr) ; KK — 2«nr. sin. t, cos. 6, sin. v. cos. -a. — ) + 3w (ar*. sin. '< sin. 'w.-r-j- ^ , . . • ''u . • , ■ d^^i + znr'te. sin.* 0 sin. wi cos. «t. «r*. sin. 6. cos. J. sin. w. cos. -a.—. — dt df- 4> 2*r' n. sin. <• cos. S, sin.'w.-; »r. sin. '*. sin. w. cos. w.- — |-2«n»'. sin.'J.sin.'iiT. —V ^ rf< dt^ ' «?// (r+(»i. )sin.(«+<»tt). sin. ( n< + OT -|-«'u)= (r + <«.)sin. <. sin.(n<-}-w + «i)) + ««r. COS.*. sin.(r!< + OT -t-«v)=r. sin. 6. sin.(n<+w)+r. sin. *. cos,(nt-\-'a).ccv-\-*ur, cos. *. sin.(ni-f-w) *s. sin.(*+«M). sin.{n<+iir-J-«v)(=«,r. sin. tf, sin.(w<+'K;).) =:r. sin, 6. sin. ■w-j-n^r. sin. 6. cos, w -j" »■• sin. *. cos. •ra-. «ii — r. sin. 6. sin. w. wt. »v ... . . dz -Hwwr. cos. sin. tf, sin.«i+«Kr. cos. «. cos.«r.Rf4-«f. sin. 6. sm.«r+«i.6in. J.cos.w.nf. .*. -r^ ' ' at ■ . I • • A) . . . . dv nr. sin. «. cos. 'o+r. sm, ^. cos. -a.u. r. sin. 6. sm. w. n«o — r. sin. ^. sm. 'a.nt».-r dt dt I . <^M du . . tfs -|-«r. cos. 0, sm. «. — -J-«Mr. cos. S. cos. w. n+«r.ni. cos. 6. cos. w.-^ + «. sin. I. sm. w.— -— uc at at . . rfs d*z . d^v + «. sin. t. cos. w. Ms-i"* sm. tf. cos. w, w/. -;— ; , , =r«. sm. «. cos. w. -rr dt dt^ af- . , . dv . . dv . d^u . du — nret.sm.6,siD,t!T.- nrx.sm.e.sin.'Br,-, — h«r. cos. 6. sin. m.—r-U anr. cos. *. cos. w.-^- , . du , . . d*s . ds . ds -t-tinr. cos. 6. cos. w. -r+a. sm. 6. sin. w. r — |-««.sm. tf. cos.w.^^ — |-«w.sin,«. cos. tr.— ; 3«=3r. sin. 6. sin. 's-f^^* >■• cos. (. sin. to+Sit. r. sin. 6. cos. w, neglecting those terms which mvolve », (at as was before mentioned, in the product 350 CELESTIAL MECHANICS, (<rr) being the value of (JF which corresponds to this state. Let us suppose that the fluid in question, is the sea j the variation(<rr) will be the product of the gravity mlutiplied, into the element of its direction. Let g represent d^z ... oz. —fT^i these quantities would produce terms of the order «, » and would consequently be df df- neglected. •.• «. — — {d^v . a . a dv , . . o d^u ru. sin. ^}. sin. «. cos. '^■j^ — 2nr». sin.'^^. sin. V.-^-{-«r. sm. fl.cos. t, sin. "w.^-j du . „ . d^s „ . . . ds\ +2«»r. sin. 6. cos. ». sin. w. cos. in.j +«. sm. '6. sin. 2©.— +2««. sm.««. sm. w. cos. «.— j- +3*. ( r'«sm. «. COS.*. sm.iff. cos. w.-tj — 2n/^<>e. sm. #• cos. <• sm. *ot.— +«r'. cos. 6. sm.^u.-^ rfa . » rf^i . . ds\ +2«Kr2.cos.2*. sin.w.cos.w.-T+aj-. sin.«. cos. 6. sin.'^cj.r-j-f 2«nr. sm.«.cos.«.sm. ro. cos.w.-^^ j- f o d% o o dv a . . <i"« J«r. •{ »■*«. sin. *<. COS. 2w. jj— 2wr*«. sm. '#. sin. w. cos. «r. — +«?*. sm. (. cos. ». em. «. cos. -cr. -^ + 2«n>-2. sin.«. cos. «. cos.^n-.^-far. sin. ^(. sin. ts-. cos. sr. -*+2«rtr.sin.-<. cos.^w. ^| , d^x , . d^y , ^ d^z d^u ,dh . ,. . .„ d-v — r«. sin. 6. cos. «. -72+«- cos. \ ^— **■• ^'^•°^' sin. ^r. cos. w.^ dv d'U . . du —2nr». sinsi. cos.^w. — + ctr. sin. 6. cos. «. cos. ^a. j-^ — 2ixrn.sm. «.cos. 6. sm. tr. cos.w. ^ +«. sin. *tf. cos. *ar. -Tj — 2««. sm. *«. sm. w. cos. w. ^ rf^t) rfu , . ■ J, d'u + ra. sin. »^. sin. v. cos. cr. jj 2nrit. sin. i'^. sin.'ar.^ +«r. sm. 6. cos. «. sin. '"--^fr +2«nr.5ia.<.cos.«,8Jn.«r.cos.zj-.^'+«. eio, *«. sin. *sr. ^+2«k. sin.'«. sin. w. cos-s-. -^ J^ PARTI.— BOOK I. 251 the force of gravity, and my the elevation of a molecule of water at its surface, above the surface of equilibrium, which surface we shall con- sider as the true level of the sea. The variation (JF) in the state of motion, will in consequence of this elevation, be increased by the quan- kk2 (= by concinnating a'ir. ( Yt — 2n»-. sin- ^^- ^) ) ;+3^- -j »^«- "n. *«.-^— r«. sin. «. cos. «. ^ d^v „ . . „ dv — <»r.* sin. «. COS. (. sin. w. cos. sr. -r- — 2nr*{c. sin. l, cos. 6. cos. V.-r— 2«r*n COS. *i. sin.o. cos. a-. — +«r.Bm.*,cos.*. cos.**— w dt dt ' rft* ds . . . d^v — 2icnr. sm. ^.cos. t, sm.«r.cos.i7.-7--|-r'« sin. i, cos. i, sin. tt. cos. 17. -rs- — ^2«r*« sin. *. cos. «. sin. ^ct. r+«'"'' cos.'^. sin. "w. —. +2«n7*.cos.*^.6in.a-.c09.w.— p at at* at . a ^'s . ^ . . ds \ -j-«r. sin. (. cos. 4. $m. V. -^ — l-2«»r*sin. tf. xos.0. S1B.0. co8.n-. -r r ' dt* dt i (and by concinnating we obtain the coe£5cient of li = to (<Pa dv \ r*tc. j-j 2n/-*« sin. «. cos. t. —J ; {d ti dv d^u. ttr*. sin. *<. sin. '«. -^^ — l-2nr*<t. sin. **.8in. w.cos. vr.-T-^xr*. sin, #. COS. t. sin.w.cos. ■w.-ry di^ ' dt d^ -♦-2«r*n. sin. ^. cos. I. sin. •w. -7 »r. sin. '<. sin. w. cos. iB.-rr+Zanr. sin. 'tfsin.Sw.-T- dt dt* dt d^v dv d^ -\-r»*.sva.*t. COS. 'w ^j 2nr*<t. sin. ««. sin. w.cos. w. ^ +*'"*• ^"^ *• cos. *. sin. «. coSj w.^ ^-2«»r*, sin. #. cos. e. cos. ^lir.^ +«r sin. »<, sin. w. cos. -an +2itnr. sin.'f.eoa'irJL 252 CELESTIAL MECHANICS, tity ^ »g-Sy ; becanse the gravity is very nearly in the direction of ay, and tends toxvards its origin ; * consequently, if we denote by a,iV', the part of SV relative to the nev? forces, which in the state of motion concinnating as before we obtain . / o . , d'v , „ , . du , „ . , ds , SzrA a^'^sin. '$. f-2«r^«, sin. (. cos. 6.—+2si}ir. sin. 6.— , }■ ^ dl- ' dt ' dt r the body having a rotatory motion about an axis, the part of the equation (H) which cor- responds to the centrifugal force arising from the rotation is by the preceding number equal to «*^3y+z32)= —. S. (_5^*-fz' ) = ^ . S. -j (r+ as), sin. («+«m) J- .'. the second num- bers of the preceding equations, when concinnated, give the equation (Z) of the text. * At the surface of the spheroid r = 1 + y ?, in which / is for simplicity^ considered as a function of i only, and the semi-axis minor= 1, .•. "ir^^q. { ^ \aw,9 depends on the eccentricity, r re- ceiving at the surface of the solid the incremented^, the corresponding increment of «=«!<» therefore the expression forr vrill become 1 -\-ql-\-ciiiq. ( j- J .'.tcs=aug.(j j and q being verj- small, s may neglected in comparison of a, and it is evidently of the order uq, i, e, of a multiplied into the eccentricity, and if/ be considered as a function of jj- only, we might shew that w receiving an increment »t), the corresponding increment of r, is to nv, as the eccentricity multiplied into I T— jis to unity. If we produce the radius r to the surface of the fluid in equilibrio ; it will be represented by l-fy l+y, y being the depth of the fluid, and a function of 6 and v, .'. 6 receiving the increment xu, the corresponding increase of the radius, dra^vn to the surface of the fluid supposed in equilibrio, will heq.( — \ .xu + (-7-) ■«!<;when the fluid is in motion, the distance of the exterior surface from thecenire,=r'-}-ai', is greater than the distance of the surface of equilibrium, from the centre of the spheroid, measured on the same radius, this last distance =,+, !+,+.». (,.( J) + (^J) ) +„. (,.(£) + (* )), ,^ W-1+? (+V+ -- ■■-('■-Wa)+(t)+-'(l)+(s))=-» = the elevation of a molecule of water in the state of motion, above the surface of PART I.—BOOK I. 25S agitate the molecule, and which arise either from the changes, which in the state of motion the attractions of the fluid and spheroid experience, or from the attractions of extraneous bodies ; we shall have at the surface, SV={SV)—ocg.Sy + x.sv: The variation — .J.{(r4-«s). sin.(fi +«")]* is increased by the quantity an* Jy.r. sin. H, * in consequence of the elevation of the molecule of the water, above the level of the sea ; but this quantity may be neglected in comparison of the term — a.g.Si/, because the ratio — ^ of the centrifugal force at the equator, to the gravity, is a very small fraction equal to . Finally, the radius r is very nearly constant at the surface of the sea, because it differs very little from a spherical surface j therefore we may make Sr=0. The equation (L) will thus, become, at the surface of the sea. r\, + T*.h {s«.{^| + .„..„...co...|^|| + >....|||| equilibrium; it is evidently a function of «andjr. y being the eccentricity, it is evident that the differential of the normal according to wliich the gravity acts, in case of equilibrium, differs from the differential of the radius, by a quantity which r: the product of the eccentricity into the differential of N, a function of 0. .-. at the surfoce of the fluid in equilibrio, {^V)—g. S, (r'+q. N), at the surface of the fluid in motion, the normal corresponding to r' -f «^, has not the same direction as when in equihbrio, its variation=S. (r'-^/jN+ai/ +«7. qN) ; the attraction of the spheroid in motion differs from the attraction of the spheroid in equilibrio by quantities of the order «^ •.• let it be equal to ai/g', then {g+^yg')- i(r'-\-qN+ui/-\-cci/ q. ]S')—(g + uyg). S[j-+jiV) + g. Uy, rejecting quantities of the order « S and remarking that ^r is of the order q.h, the first term of the second member of the preceding equationz=(S f^ . •. the second term is the quantity by which in the slate of motion (S V) is increased, as has been stated in the text. 254 CELESTIAL MECHANICS, the variations Sy, and SV, being taken relatively to the two variables 6, and w. Let us now, consider, the equation relative to the continuity of the fluid. For this purpose, let us conceive at the origin of the motion, a rectangular parallelepiped, of which the altitude is dr, the breadth r. dw. sin. fi. and the length 7-.d9. * Let r', 8', ir', represent what r, 6, ■a-, become after the time /. By following the reasoning of No. 32, we shall find that after this interval, the volume of the molecule of the fluid, is equal to a rectangular paral- (dr't lelepiped, of which the height is -j — r- .dr ; of which the breadth is dr being eliminated, by means of the equation Finally, its length is ' • r Bin. » = radius of a smaU circle, whose plane is parallel to the equator, and as the plane of the axes of x, and y, is fixed, r, sin. 6. dw= the differential of the arc of thi« circle, to wliich dr is evidently perpendicular, also, the differential of the nieridian:=r.<^«, is perpendicular both to r. «aa. i. d«r and to dr, .•. these three differentials, constitute the parallelepiped mentioned in the text. • When the fluid is in motion, this expression becomes, — J.(r+i»4 + <»y) sin. (« + »«)* .-.the part which corresponds to«y, is «*.«3y.(r-J-<M-f-<«y). sin.(»+<tu)» = by neglecting quantities of the order «*, tv'.aii/. r. sin. \ PART I.— BOOK I. S55 dr, and iv, being eliminated by means of the equations Consequently, if we make (dr'X (d^'X f dzr' \ (dr'X ( d^l ( rfsr' | ^= \dP yid^yx 'd^yX'^rfXd^l '{w j jd/\ (d^\ (d£X after the time t, the volume of the parallelepiped will be equal to C. r*. siu. 6. dr. d9. d^- ; * therefore if (p) represent the primitive density of the molecule, and /> its density, con-esponding to the time t, we shall obtain, by putting the primitive value of its mass, equal to its value after the time t, p. e'r'*. sin. y = (p). r*. sin. 9 ; this is the equation relative to the continuity of the fluid. In the case we are at present considering, r' = r+a.s; 6' = 9 + «u; ■ar=nt+zr + aV', * r" t ■s/ are generally functions of r, t, w, and t, see page 217, notes ; the reasoning is precisely the same as in page 218, substituting the coordinates j-, t, w, in place of j;. ;/> *• 256 CELESTIAL MECHANICS, consequently, we shall have by neglecting quantities of the order «• Let us suppose that after the time /, the primitive density (p) is changed into (p) + «p' ; the preceding equation relative to the continuity of the fluid, will give 36. Let us apply these results, to the oscillations of the sea. Its mass being homogeneous, f' vanishes, consequently, » dr'=dr->r»ds, d^=d6-^»du,d^*d^J^*dv .'. (x-)-(^)-(^)=-- (dr-\-ctds\ (d6-^»du\ ^f dtn-X-adv \ , . {" ds \ , / du\ /dv \ it is plain, that if there was no motion, the differential of any coordinate 6, with respect to another coordinate, would vanish, after the time <, this differential is of the order <••. — -7— — ^ ^ ( ^T** — )'s of the order f or «■", consequently it may be neglected, from (JOT ) ^ di ' which it appears, that all the terms in expression for €' after the first may be neglected. r»{sin. «+«M.cosO \ =(5). r». sin. «) (»''+2«s). sin. <+«M. cos, i) > =(§). r*. sin. t, i, e, PART I.— BOOK I. 257 Let us suppose, conformably to what appears to be the case of nature, that the depth of the sea is very small in comparison with the radius r of the terrestrial spheroid; let this depth be represented by y, y being a very small function of 6 and -a, which depends on the law of this depth. If we inte- grate the preceding differential equation, with respect to r, from the surface of the solid which the sea covers, to the surface of the sea, * it is obvious that the value of 5 will be equal to a function of 9, w, and /, independent of LL (j). (r»-l-2«r5).(sln. ()+««cos. 0)+ («)• '■'•«n. i. J«-{§} + {^] + {£} } +aj'. r'. sin. 6-=z{() r«. sin. 6. .-. (j).r«.««cos.«+(5)2*w.sin. «+(j).r'.sin.«.|«|^|+ {^|+ {^}} -}-«§'. r'. sin, tf=0 .•. dividing by sin. 6 and », we obtain • The depth of the sea being inconsiderable, in comparison of the terrestrial radius, we may suppose, that for this depth r*, and the factor of r» in the second tema, of the second member of this equation, are constant .*. integrating we obtain , / /du\ , tdv \ u COS. (■y as the increment of the radius at the surface of the spheroid = aug. (-j-j +avg. ( j- ) see notes to page 252, .-. s' at the surface of the sea _ f (du-) Cdvl tt.cos. «1 . <dll , / dl\ €58 CELESTIAL MECHANICS, r,togethev with a very small funetion which will be to u and tof , of the same order of smallness as the function _; but at the surface of the solid which r tiljie sea covers, when the anglesfi, and^D-, are resj^ectively changed into 6 + a«, v-^nt-^ «i', it is easy to perceive that the distance of a molecule of water, contiguous to this surface, from the centre of gravity of the earth, only varies by a quantity very small with respect to a.u and a,v, and of the same order, as the products of these quantities, into the eccentricity of the spheroid covered by the sea : therefore, the function, which occurs in the expression for s, independent of the value of r, is a very small quan- tity of the same order ; thus we can generally neglect s, as inconsiderable, in comparison of u and v. Consequently, the equation of the motion of the sea, which has been given in No. Z5, becomes, + r*Sz7Ss,{n.^L\~l + 2n. sin. 6. cos. 9. i — } |- =—g.S^+SV'; (M) the equation (L) of the same number relative to any point of the interior of the fluid, gives in the state of equilibrium, 0- I'. J. ( (r + a5). sin (9 + cu) \ + {SV) — ^-^ (iV)- and {Sp) being the values of iV and S<p^ which in the state of equi- these two last terms are to u, or v, as the product of these quantities into the eccentricity. With respect to the first term, it may be remarked that we can derive another expression for it, in terms of the difference of the eccentricities of the interior and exterior spheroids, divided by r, but tlus difference is evidently proportional to y, in fact this term will be to ur as y to r. The integral involves ( because it was taken with respect to tbe characteiistic d and not 3. The last member of the equation ( L ) bcconjes in a, state of motion, ia consequence of Uiig substitution, PART I.— BOOK L ^ft librium, answer to the quantities r+ocs, 6 + *m, u + ttt. Suppose thit when the fluid is in motion, we have the equation (L) will give From a consideration of the equation (M), it appears that "•! j7 I ^^ ^'^ the same order as 1/ or s, and consequently of the order — ; the value of the first member of this equation is therefore of the same order ; * thus, multiplying this value, by dr, and then integrating from the surface of the spheroid, to the surface of the sea; we shall have V — ^equal to a very small function, of the order-i-, plus a function of 9,Tr, and t, independent of r, which we will denote by x; therefore, if in the ♦ |!. J S^ (r+«M).6m.(*+«tt) I *+(>r)— {- } + «3^'— « — . the three first terms destroy each other .•. aiV — « — +| is equal to the first member of the equation (L), and i 1/ liace it is an exact variation, the first member of the equation(L)wiU Be so also, .*. V — dif- ftfenced With respect f 0 f, is equal M the tentt of the first metrHbex of the equatioB (t). wWch is multipKed by Sf . — «. sin.«.cos. »= t.'^cos.<0 , in order that —2». ^ ^ ^ . sin. <. cos. » may b* *f ifienme wfder m | - - I it 16 Becessay that « | — ? should be of the order y or </whi«K is of the order — 260 CELESTIAL MECHANICS, equation (L) of No. 35, we only consider the two variables G and zr, it will be changed into the equation (M), with this sole difference, that the second member will be changed into <5'x. But A being independent of the depth of the molecule, which we consider; if we suppose this molecule very near the surface ; the equation (L) must evidently coincide with the equation (M) ; therefore we have SxzzSV — gSy, and con- sequently, S.\v'^^^'^=SV'^gSy; the value oiSV'in the second member of this equation, being relative to the surface of the sea.* We shall find in the theory of the flux and reflux of the sea, that this value is very nearly the same for all molecules situated on the same terrestrial radius, from the surface of the solid which the sea covers, to the surface of the sea; therefore with respect to all thesemolecules Sp' . . — zzg.Sy; which gives ^' = f^T/, together with a function independent of 6, T3-, and r ; but at the surface of the level of the sea, the value of a.p', is equal to the pressure of a small column of water uy, which is elevated *J j-^dr—'2/nrdr.sm.'i6 < — \ integrated between the the siuface of the splieroid, and ( d''s 7 the surface of sea, gives the integral of the text, the first term is — to i - — > y, which is a function of ^.•ct, and <,=A, the other term being of the order —may be rejected. If we only consider the terms, which refer to S and -r, the first member of the equation (L) is the same as the first member of the equation (M), near the surface, the last term of the first member of the equation (L') vanishes .-. the equation (L) must in this case coincide with the equation (M), but A the member of the equation (L) does not vary .•. we have the second member of the equation (L) — the second member of the equation(M) i. e, Ja=JF' — giy '■> but Sa = 3./ V'—JL I ••• S- •[ V'— ^ ? =.lV'—g.ly, from the theory of the tides * / it appears that the 5 F' fin these two members are the same, .•. g^xf=:.— and ■p'^egy -^ a con- % slant arbitrary quantity ; when the integral is taken between the surface of spheroid, and eurface of the sea, this constant arbitrary quantity may be rejected. PART I.— BOOK I. 261 above this surface, and this pressure is equal to a-^.gy % therefore we have, in the entire of the interior of the fluid, from the surface of the spheroid covered by the sea, to the surface of the level of the sea, p' = ^gy ; conse- quently, any point of the surface of the spheroid, which is covered by the sea, is more pressed than in the state of equilibrium, by the entire weight of a column of water, contained between the surface of the sea, and the sur- face of level. This excess of pressure becomes negative, for those points, where the surface of the sea is depressed beneath the surface of level. It follows from which has been stated above, that if we only consider the variations of 9 and is ; the equation f L) will be changed into the equation (Mj, for all the interior molecules of the fluid. Consequently, the values of u, and v, relative to all molecules, * situated on the same terrestrial radius, are determined by the same differential equations ; thus, supposing, as we shall do in the theory of the flux and reflux of the sea, that at the commencement ofthemotion,the values of w,i — Vy, I — I, were the same for all the molecules of the fluid, situated on the sanie radius, these molecules will exist the same radius, during the oscillations of the fluid. Therefore the values of r, u, and v, may be supposed very nearly the same, on the small part of the radius, comprised between the solid, which the sea covers, and the surface of the sea ; thus, if we integrate with respect to r, the equation ^ cd.r*s 1 , , ^^du) , (dv} , u cos 9 } , * At the commencement of the motion u, and v, i — f i "{ t; r > ^^^ the same, for all molecules situated on the same radius, .•• after the interval dt, the corresponding values of u and V, will be the same for all molecules situated on the same radius. t r-s-(j^i)zz)%_?2.(^)4-2,-y.(i)+y2(i)for (r^)=(r—y)* y being a function of i, and ar, when these angles are increased by the quantity «m, »v, becomes y-\-aM.. \-t-\ +«d. 5 j - f this is the value of -/ con-esponding to the angle <+«u, ^■\-nt-\-an for the surface of equilibrium, ,•• where the fluid is in motion, we must add ay to this expression. COS. 6 262 CELESTIAL MECHANICS, we shall have ( crf9 J Ccra-J sm. 8 ) (r»5) being the value of Vs, at the surface of the spheroid covered by the sea. The function r^s — (r*s) is very nearly equal to r*. [s — («)} +Qry(s), (s) being what 5 becomes at the surface of the spheroid ; con- sidering, the smallness of y, and (5), in comparison of r, we may neglect the term 2ry.(s) ; therefore, we shall have ros—(r*s')=r.' [«— (5)}. Now, the depth of the sea, corresponding to the angles 0-|-ixw, ar + w?+«f, is y + a.[s — (s)]. If the origin of the angles 6, and ni + sr, be referred to a point, and a meridian, which are fixed on the surface of the earth, which we are permitted to do, as we shall see very soon ; this same depth will be y-i- au. ^-T^^+ai'. j;r-(> plus the elevation ay of the molecule of the fluid at the surface of the sea, above the surface of level ; therefore, we shall have If we make cos. 4=:^, then sin. « ' '^ ^•' 1 — u^ —dfi ^ ^. __ —fi.dft. ^ __j^_ COS. i sin. $ consequently the equation of continuity, on the supposition that the sea is honiogeneouA becomes, — (^) +r2. r ^^ _r2. (rf.(«-v/rV)w Book IV. Chap. 2. ^ Md^ r+ :?: — +-d;;^ j= -r- { [^)+ —T^ S.ee Book IV. Chap. 1, No. 2. PART I.— BOOK I. 263 '-»=^+-gi( + 4l}' Consequently the equation relative to the continuity of the fluid will become * cdyjO (d.yV? yii.cos.6 ,„. It may be remarked, that in this equation, the angles 8 and nt-\-t!r are reckoned from a point, and a meridian, which are respectively fixed on the surface of the earth,and in the equation(M), these angles are reckoned from the axis of a, and from a plane, which passing through this axis, revolves about it with a rotatory motion, expressed by n ; but this axis, and this plane are not fixed on the surface of the earth, since the attraction and pressure of the fluid which covers it, as well as the rotatory motion of the spheroid, disturb a little their position. However it is easy to perceive that these perturbations t are to the values of «m, and «t', in the ratio of the mass * Substituting for s — {s), its value {du "i dv di > ^' 1Z and observing that du ^ dv lu COS. * y. . sm. i we will arrive at the value of y, which is given in the text. f In the state of equilibrium, neither the pressure or attraction of the ocean, can produce any motion in the spheroid covered by the sea, and it is only the stratum of water which irv consequence of the attractions of the exterior bodies, and of the centrifugal force, is elevated above the surface, wliich can produce any effect. The effects of the pressure and attraction^ may be considered separately, with respect to the first, if the mean radius of the earth be supposed equal to unity, «y being the elevation, the action of the aqueous stratum is equal to the diiference of the attractions of two spheroids, of which the radius of the interiors 1, of the exterior — l+»i/, naming this difference »y.k. and t its direction, uyhdv will be the expression for this attraction ; multiplied into the element of its direction, t being a function of <, and -a, dr 26i CELESTIAL MECHANICS, of the sea, to the mass of the spheroid ; therefore, in order to refer the angles 9, and nt+zi; to a point and meridian, which are invariable on the surface of the spheroid, in the two equations (M) and (N) ; we should alter u, andi^, by quantities of tl:e order^ and — , which quantities we r r are permitted to neglect ; therefore we may suppose in these equations, that a.u and a.v are the motions of the fluid, in latitude and longitude.* It may also be observed, that the centre of gravity of the spheroid being supposed immoveable, we should transfer in an opposite direction to the molecules, the forces by which it is actuated, in consequence of the re- action of the sea ; but the common centre of gravity of the sea and sphe- roid being invariable in consequence of this reaction ; it is manifest that the ratio of these forces, to those by which the molecules are solicited by the action of the spheroid, is of the same order, as the ratio of the mass therefore they may be omitted in the calculation of W. of the fluid to that of the spheroid, and consequently of the order-, The attractions are of the order ay ; for if y vanished there would be no pressure or action, but y is of the order — . The exact effect which the attractions, and pressures of the aqueous stratum produce are calculated in Book V. Nos. 10 and 11. • The centre of gravity of the spheroid is considered immoveable, because we do not consider the absolute oscillations of the molecules in space, but only their oscillations reia^ live to the mass of the fluid. The common centre of gravity of the fluid and spheroid covered by the fluid is not affected by the mutual action of these molecules, see No. 20. With respect to the action of foreign bodies, their effect is not to be neglected, as in case of the action of the sea, if we consider the centre of gravity of the spheroid immoveable, we must transfer in a contrary direction to the molecule, the attraction which such bodies exert on the centre of gravity of the spheroid, the oscillations «y and the force which actuates the particles are of the order a.-~fix ».q. \ — -i- !• ,see preceding note. PART L— BOOK I. 265 37. Let us consider in the same manner, the motions of the atmos- phere. In this investigation, we shall omit the consideration of the variation of heat in different latitudes, and different elevations, as well as all anomalous causes of perturbation, and consider only the regular causes which act upon it, as upon the ocean. Consequently, we may con- sider the sea as surrounded by an elastic fluid of an uniform temperature ; we shall also suppose, that the density of this fluid is proportional to its pressure, which is conformable to experience. This supposition implies, * that the atmosphere has an infinite height ; but it is easy to be assured, that at a very small height, its density is so small, that it may be regarded as evanescent. This being premised, let s', u\ and w', denote for the molecules of the atmosphere, what s, u x\ designated, for the molecules of the sea ; the equation (L) of No. 35, will then become 2 .„ t \d*ii {m--^->--'-m\ + 1 ^ 3- ( • 2fl /■ d""^' \ . o -ft „ A fdu'\ , 2w. sin, ^9 / ds' \\ '\-o^'rJzT.\sm.^\—-^) +271. sin. 9. cos. 9, ( — 1 -| .( — ) ( dt- ' ^dt ' r ^ dt ' . <^-Sr. $ (^~)—2nr. sin. \ (^)^ = |-. <^.(^+ ccs').sm.^ + xu').Y +^F- ^P. e M M • x\ccording as the fluid is elevated above the surface of the earth, it becomes rarer, in consequence of its elasticity which dilates it more and more, as it is less compressed, and it would extend indefinitely, and eventually dissipate itself in space, if the molecules of its surface were elastic ; consequently, if there is a state of rarity, in which the molecules are devoid of elasticity, the elasticity of the atmosphere must diminish in a greater ratio than the compressing force. 266 CELESTIAL MECHANICS, At first let us consider the atmosphere in a state of equilibrium* in which case s\ zi' andt/ vanish. Then, the preceding equation, being integrated becomes, ■— .r^. sin. *9 + F— P-^ = constant. The pressure p being by hypothesis proportional to the density ; we shall make j) = I. g. p, g represents the gravity at a determined place, * which we will suppose to be the equator, and / is a constaat quantity which expresses the height of the atmosphere, of which the density is throughout the same as at the surface of the sea : this height is very small relative to the radius of the terrestrial spheroid, of wliich it is less than the 72Dth part. The integral A^ is equal to Ig. log. f ; consequently the preceding equation relative to the equilibrium of the atmosphere becomes, ig. log. p = constant + r+ — .-r*. -sin. *9. At the surface of the sea, the value of F" is -the same for a molecule of air, as for a molecule of water contiguous to it, because the forces which solicit each molecule, are the same ; but the condition of the equilibrium of the sea requires, that wc should have V-)r — . r^. sin.^S=constant ; 2 * An homogeneous atmosphere is an atmosphere, supposed to be of the same weight as that which actually surrounds the earth ; its density being uniform, and every where equal to the density of the air at the surface of the earth. Let h be the height of the mercury in the barometer at the equator, and d its density, we shall have lg=h.d:. /x— and by e substituting for A and e^ and g their rnumerical values, /comes out equal to 5;^ miles very nearly, which is somewhat less than ^he 720th part of the radius of the equator. When the temperature is given, this height is a constant quantity, whatever be the ohang«s wliich 'the pressure undergoes. PART I— BOOK I. «67 therefore p is constant at this surface, i, e, the density of the stratum of air contiguous to the sea, is every where the same, in the state of equilibrium. Let R represent, the part of the radius r, comprehended between the centte of the spheroid and the surface of the sea, and r' the part comprised between this surface and a molecule of air ele- vated above it ; r' will differ only by quantities nearly of the order — . r' 1 , * from the /leigkt of this molecule above the surface of the sea ; we may without sensible error neglect quantities of this order. The equation between p and r will give Ig. log. f = constant + l + — . RK sin. *H?^* -R/. sin. -fl : 2 the values of V, (-p^and (-r-g) being relative to the surface of the sea, where we have, constant = V+ ^'R'- sin. -9j the quantity *- l—^ \— n* R, sin. % expresses the gravity at the same M m2 * V being a function of R, 6, and vt, i{ R receive the increment /, V becomes s V "^ T I d~ \ "^ T9\ J~t\'^ ^^' ^^ the expression^ R^- sin. -e will be increased by the quantity n* R/, sin. *«^ — -/.' sin. *(, but this last term being indefinitely small, may be rejected. 268 CELESTIAL MECHANICS, surface; which we will represent by g'. The function \tll * being niul^ !;/:> ■€. Ldr 3 ^ tiphed by a very small quantity r? we may determine it on the hypothesis, that the earth i. spherical, and we may neglect the density of the atmos- phere relatively to that of the earth ; therefore, we shall have very nearly, Ur i ~ * m' m expressing the mass of the earth ; consequently S— J = '~Jlz= — ^ ; therefore we shall have /^. log. p= constant —^'g' — ^g' ; from which may be obtained _r'g' C r) p = n,c t *If the earth was a sphere then r', would be equal to the height of the molecule of the atmos* phere above the surface of the sea, and as in the case of a spheroid the height is determined by a normal drawn to the surface from the molecule, the difference between / and the part of this normal which is exterior to the surface, depends on the ellipticity of the spheroid, which is 1 for he afterwards supposes that the earth is at the surface of the sea very nearly ^ spherical, .*. the only abberration from sphericity can arise from the greater centrifugal force of the molecule of the air, the ratio of this excess of cen- trifugal force to gravity, for a molecule elevated at the equator, above the surface of the earth r= , and the mtercept at the surface between the du-ection of r, and the direction of a normal drawn from the molecule of the air must be evidently of the order of the ellipticity t, e, of the order , and the difference between r' and this height is equal to the square of this quantity divided by R very nearly. t Sf— P3x+ Qlj/-i-Rh, and if we refer the molecules to the polar coordinates r, i, w. PART I.— BOOK I. . 269 c being the number of which the hyperbolical logarithm is equal to unity, and n being a constant quantity evidently equal to the density of the air at the surface of the sea. Let h and /?' represent the lengths of a pen- dulum, which vibrates seconds at the surface of sea, under the equator, and at the latitude of the molecule of the atmosphere, which has been is that part of the force SF, which is resolved in the direction of the radius of the earth, tf:= the complement of latitude .'. »*7l sin. '< is the part of the centrifugal force, which acts in the direction of the terrestrial radius. The force varying inversely as the square of the dis- 1 dV m tance, V-^ —, and — r: -j- see Book II. No. 12. R dr R' The earth being supposed spherical 5 ;7- >• 'S nearly the same in every parallel, and .-. equal (d'Vt to its value at the equator, where it is equal to g very nearly ; in the value of < -j-^ > we sub- stitute ^ in place of ^^ , for thus the error of the supposition that g =:^is somewhat cor- rected ; substituting for /'dV\ „ . . fd^V\ «» „, . (^-) + mR.sm.^>,^^)+-^R'.sm.'e .their values and of remarjiing that V-{- — R.* sin. '« is constant, we obtain the value of Ig. log. ^ which is given in the text. The density of the atmosphere being inconsiderable with respect to that of the earth, we may without sensible error, neglect the attraction of its molecules. The variable part of the value of § is necessarily negative, for the density decreases, ac- cording as we ascend in the atmosphere ; const ^g'-(.y\ Ig Ig V-^Rl , const r'c^.,1 , r'\ const md at the surface of the sea / :rO .••{=<: = n which is consequently the value of 5 at the surface of the sea; when the times of vibration are given, the lengths of the isochronous lendulums are proportional to the forces of gravity, .*. — :i -r-. 270 CELESTIAL MECHANICS, considered : we shall have— = -, and consequently, g h 7/ C r'\ Ih f—n. c * From this expression of the density of the air, it appears that strata of the same density, are throughout equally elevated about the surface of the sea, with the exception of the quantity -i — ~-^ j however, in the exact determination of the heights of mountains by observations of the barometer, this quantity ought not to be neglected. Let us now consider the atmosphere in a state of motion, and let the oscillations of a stratum of level, or of the same density in the state of equilibrium, be determined. Let acp represent the elevation of a mole- cule of the fluid, above the surface of level, to which it appertains in the * If we expand the value of g into a series it becomes equal to ■ V ' r'h' and neglecting higher powers of /',=!— =7- .-. in strata of equal elevation above the level h'—h of the sea, the difference of density is equal to r. (j — j ; in like manner, if the density of two strata, in latitudes of which the forces are respectively equal tog and g'; be the same, we shall have Ih 7' and r* being the heights which coiTespond to the respective latitudes, .•. neglecting quantities of the second order we shall have, when the density is given, /A'=r"A .-. r'/= — conse- h quently the difference between/ and /''/(=—)= r'. ('— ^V PARTI.— BOOK I. 271 state of equilibrium ; it is manifest that, in consequence of this eleva- tion, the value of tVvfill be increased by the differential variation —»g.S<p ; thus we shall have, SV:=.{iV)-^s>i.g.Sip + »SV' ; (^SV) being the value of S V, which, in the state of equilibrium, corresponds to the stratum of level, and to the angles 9 + «w, and nt+zr+otv j SV being the part of iV, which is produced by the new forces, which in the state of motion, agitate the atmosphere. Let fi=:(f) + «f', f being the density of the stratum of level, in the state of equilibrium. By making -4-=y, we shall have but in the state of equilibrium we have, 0= ^J.{{r+»s). sin. ($ + «^)}»+(JD-/^'y } therefore, the general equation relative to the motion of the atmosphere will become, relatively to the strata of level, with respect to which ir very nearly vanishes, +r^.J^.|sm. .9. |-^j+2«. sin. 6. cos. 6.^-^1+ r-\rf^|| = neglecting quantities of tlie order a', Ig- 8(5) (•(s)~(g).~-«.{') 272 CELESTIAL MECHANICS, =iV'—gJ(P'^gSy' + n^r. sin. »6.$. (/ — (/)),* a («') being the variation of r, which in the state of equilibrium corre- sponds to the variations a?/, «u', of the angles 9, and zy. Let us suppose that all the molecules, which at the commencement of the motion existed on the same radius vector, remained constantly on the same radius in a state of motion, which, as appears from what pre- cedes, obtains in the oscillations of the sea ; and let us examine whether this supposition is consistent with the equations of the motion and continuity of the atmospjiere. For this purpose, it is necessary that the values of u' and of v', should be the same for all these molecules, as we shall see in the sequel, when the forces which cause this variation are de- termined ; consequently, it is necessary that the variations Sip and St/, should be the same for these molecules, and moreover that the quantities •V 2nr. S-sr. sin. 'Q. S'^> , and n'r. sln.^^ J. S <t' — (s) (, may be neglected in the preceding equation. At the surface of the sea, we have <p=]/, a-y being the elevation of the surface of the sea above the surface of level. Let us examine whether the suppositions of <? equal to y, and of y constant for all molecules of the atmosphere, existing on the same radius vector, is compatible with the equation of the continuity of the fluid. This equation is by No. 2,5, * ai' and «(^') being tlic variations of r, corresponding respectively, in the states of motion and equilibrium, to the variations mi and «.v' , the expression p. 3. J (r-f «s'). sin.(i!-l-««') j '= Y- ^- { C*" ^■ «(*')+«(«'-(*'))• s:n-(«+««) } ' and when we neglect quantities of the order**, the part of this expression, which docs not occur in the equation 0=-|-. 3. |(»- + «4 sin.(tf+«i<) I , is, 7j« r.«.S. | (*'—(/) |. sin. '«. PART I.— BOOK I. 273 fiom which we obtain , C f d.r's' > , Sdu") ^ <:dt'} u.' cos. 9 % r + as' is equal to the value of r at the surface of level, which corresponds to the angles 8+«m, and sr + uv, together with the elevation of a molecule of air above this surface ; the part of as' which depends on the variation of the angles G and iB-jt being of the order — '—, may be neglected in & N N * Dividing tliis equation by r» (5) we shall obtain (?) ^~ I' ^d« ) W j sin. 6 \r' dr )' f The part of «/ which corresponds to the variations »u', av', is of the same order as the products of these quantities by the eccentricity of the spheroid, see page 258, and the ec- centricity in this case is proportional to the fraction — , consequently the variation of «/ which corresponds to the variation of the angles I and 1?,=: ; the entire variation of cts' is g made up of two parts, of which one is equal to the elevation of the molecule above the sur- face of equilibrium, on the supposition that the angles 6 and ar are not varied, and this part of the variation of us'=ttip, the other part of the variation is the part which corresponds to the variations au' and «u' of the angles 6 and w, and from what precedes it appears that this part may be neglected, consequently we have r d.r^sf \ 2s' . ds' the second term= f~- J by substituting <p in place of y, to which it is equal, and when (f> is supposed to be equal to y ; its derivitive function with respect to r must vanish, <p being the same for all the molecules, situated on the same radius, y is the same order &ss', or the eccen- 274 CELESTIAL MECHANICS. the preceding expression for y, consequently it may be supposed in this ex- pression thats'=:i?; by making <p =3/, we shall have! — | = 0,sincethe value of (p is then the same for all molecules situated on the same radius. Moreover, by what precedes, y is of the order Z or — ; therefore the ex- pression for y' will become. '_ 1 W*^^' \ \^^ y m'. cos. 6 ^~ '\Xdf\ Id^S sin. 9 thus, 11 and v' being the same for all molecules situated primitively on the same radius, the value of y will be the same for all these molecules. Moreover, it is manifest from what has been stated that the quantities <2nr. iw. sin. -9. j— Land «V.sin.^9. (?.(«'— .-(y)), may be neglected in the preceding equations of the motion of the at- mosphere, which can then be satisfied, by supposing that u' and t' are the same for all the molecules of the atmosphere, which at the commencement of the motion existed on the same radius ; therefore the supposition that all those molecules remain constantly on the same radius during the oscil- lations, is compatible with the equations of the motion and of the con- tinuity of the atmospheric fluid. In this case, the oscillations of the different strata of level are the same, and may be determined by means of the equations, tricity which is proportional to -, and this last quantity is proportional to I, see page 258 and o 2s' ds' 266, ••• we may neglect both — and — consequently we will obtain for 1/' the expression given in the text. It is manifest from what has been stated in notes to page 253, tliat — y«^./-sin.*^.J(i' — («))raaybeneglectedwhenthe earth is nearly spherical. PART I.—BOOK I. il5 +r.»J^.5sm.=fl.J^ I — 2n. sin. 9. cos. 9. Yj-\ -^'^'—g-^'S'^r, t_ , C Cc?m'> (cfw'? z/cos 9 ? These oscillations of the atmosphere ought to produce corresponding oscillations,, in the heights of the barometer. In order to determine these last by means of the first, we should suppose a barometer fixed at any elevation above the level of the sea. The altitude of the mercury is pro- portional to the pressure which the surface exposed to the action of the air experiences ; therefore it amy be represented by Ig. p ; but this surface is successively exposed to the action of different strata of level, which are alternately elevated and depressed like the surface of the sea ; thus the value of p at the surface of the mercury varies, 1st, * because it appertains to a stratum of level, which in the state of equilibrium was less elevated by the quantity a.y; 2dly, because the density ofa stratum increases in the state of motion, by a/ or by— yi^ . In consequence of the first cause, the variation of f is augmented by the quantity — »y, ( -f }or ^'pi. therefore the en- tire variation of the density f at the surface of the mercury, is gtCp)- , . It follows from this, that if we represent the height of the mercury, in ^ * (rfr)~ '^' (/)• '° *^ *^'® °^ equilibrium /^\=g'. (§) see (page 223) ••. (^\ = ^^ consequently — ay, ( —~ J = ' j \-r-j 's negative because the density in- creases as we ascend in the atmosphere . The temperature of the air being supposed to remain unvaried, its specific gravity will vary as ({) its density, and this quantity varies as Jc. 276 CELESTIAL MECHANICS, the barometer, in the state of equilibrium by k j its oscillations, in the state of motion will be represented by the function — '^" ; conse- quently at all heights above the level of the sea, these oscillations are similar, and proportional to the altitudes of the barometer. It only now remains, in order to determine the oscillations of the sea, and of the atmosphere, to know the forces which act on these respective fluids, and to integrate the preceding differential equations j which will be done in the sequel of this work. END OF THE FIRST BOOK. TREATISE OF CELESTIAL MECHANICS, BY P. S. LAPLACE, MEMBER OF THE NATIONAL INSTITUTE, &C. PART THE FIRST— BOOK THE SECOND. TRANSLATED FROM THE FRENCH, AND ELUCIDATED WITH EXPLANATORY NOTES. BY THE REV. HENRY H. HARTE, F.T.CD. M.R.I.A. DUBLIN : PRINTED AT THE UNIVERSITY PRESS, FOR RICHARD MILLIKEN AND HODGES AND M'AKTHUR. 1827. R. GRAISBERRy, PRINTER Ty THE O.NIVERSITT. TABLE OF CONTENTS. BOOK II. Of the law of Universal Gravitation, and of the Motion of the centre of gravity of the Heavenly Bodies. -..-.. Page 1 CHAPTER I. Of the law of Universal Gravitation, deduced from the ■phenomena. The areas described by the radii vectores of the planets, in their motion about the sun, being proportional to the times ; the force which sollicits the planets, is directed to- wards the centre of the sun, and conversely, .... No. 1 The orbits of the planets and comets being conic sections; the force which actuates them is in the inverse ratio of the square of the distance of the centres of these stars from that of the sun. Conversely, if the force varies in this ratio, the curve described is a conic section, ...... .- No. 2 The squares of the times of the revolutions of the planets being proportional to the cubes of the major axes of their orbits, or, what comes to the «ame thing, the areas described in the same time, in different orbits, being proportional to the square roots of their parameters ; the force which sollicits the planets and comets ivill be the same for all bodies placed at the same distance from the sun, ... No. 3 The satellites in their motions about their respective primary planets, present very nearly the same phenomena, as the planets do in their motion about the sun ; therefore the satellites are soUicited towards their respective primary planets and towards the sun, by forces which vary inversely as the squares of the distances. - - No. 4 Determination of the lunar parallax, by means of experiments made on heavy bodies' and on the hypothesis of the force of gravity varying inversely as the square of the distances. The result which is thus obtained, being perfectly conformable to obser- vatiofis; the attractive force of the earth is of the same nature as that of all the hea- venly bodies ........ No. 5 IV CONTENTS. General reflections on what precedes ; they lead to this principle, namely, that all the molecules of matter attract each other directly as the masses, and inversely as the square of the distances, -------. No. 6 CHAP. II. Of the differential equations of the motion of a system of bodies subject to their mutual attraction, - • - . . . 27 Differential equations of this motion. - - - - - No. 7. Development of the integrals which we have been hitherto able to obtain, and which re- sult from the principles of the conservation of the motion of the centre of gravity, of areas, and of living forces. ....... fjo. 8 Differential equations of the motion of a system of bodies, subject to their mutual at- traction, about one of them, considered as the centre of their motions ; development of the rigorous integrals which we have been able to obtain. - - No. 9 The motion of the centre of gravity of the system of a planet and of its satellites about the sun, is very nearly the same as if all the bodies of this system were united in this point; and the system acts on the other bodies, very nearly as in this hypothesis. No. 10 Discussions on the attraction of spheroids : this attraction is given by the partial dif- ferences of the function which expresses the sum of the molecules of the spheroid, divided by their distances from the attracted point. Fundamental equation of partial differences, which this function satisfies. Different transformations of this equation. No. 11 Application to the case in which the attracting body is a spherical stratum : it may be proved, that a point situated in the interior of a spherical stratum is equally attracted in every direction ; and that a point situated without the stratum is attracted by it, as if the mass was condensed in its centre. This result likewise obtains for globes com- posed of concentrical strata, of a variable density from the centre to the circum- ference. Investigation of the laws of attraction in which those properties obtain. Among the infinite number of laws which render the attraction very small at consi- derable distances, that of nature is the only one in which spheres act on an exterior point, as if their masses were united in their centres. This is likewise the only one, in which the action of a spherical stratum on a, point situated within the stratum vanishes. ' . ^ No. 12 Application of the formulae of N°. 11, tp the case in which the attracting body is a cy- linder, of which the base is a reentrant curve, and of which the length is infinite. When this cuiTe is a circle, the action of the cylinder on an exterior point is reciprocally proportional to the distance of this point from the axis of the cylinder. A point situ- ated in the interior of a circular cylindrical stratum, of a uniform thickness, is equally attracted in every direction. ...... No. 1 3 Equation of condition relative to the motion of a body. ... No* 14 CONTENTS. V Different transformations of the differential equations of the motion of a system of bodies subject to their mutual attraction. ..... No, 15 CHAP. III. First approximation of the celettial motions, or the theory of elliptic motion. - - - - ---97 Integration of the differential equations, which determine the relative motion of two bodies, attracting each other directly as the masses, and inversely as the square of the distances. The curve which they describe in this motion is a conic section. Ex- pression of the time in a converging series of the sines and cosines of the true motion. If the masses of the planets be neglected relatively to that of the sun, the squares of the times of the revolutions are as the cubes of the greater axes of the orbits. This law obtains in the case of the motion of the satellites about their respective primary planets. No. 16 Second method of integrating the differential equations of the preceding number. No. 17 Third method of integrating the same equations ; this method has the advantage of fur- nishing the arbitrary quantities, in functions of the coordinates and of their first dif- ferences. ....... Nos. 18 and 19 Finite equations of elliptic motion : expressions of the mean anomaly, of the radius vector, and of the true anomaly, in functions of the excentric anomaly. - No. 20 General method for the reduction of functions into series ; theorems which result from it. . - - - - - . - - No. 21 Application of these theorems to elliptic motion. Expressions of the excentric anomaly, of the true anomaly, and of the radius vector of the planets into converging series of the sines and cosines of the mean anomaly. Expressions in converging series, of the longitude, latitude, and of the projection of the radius vector, on a fixed plane, a little inclined to that of the orbit. ...... No. 22 Converging expressions for the radius vector and time, in functions of the true anomaly, for an extremely excentric orbit. If the orbit is parabolic, the equation between the time and true anomaly is an equation of the third degree, wliich may be solved by means of the tables lof the motion of comets. Correction which ought to be applied to the true anomaly computed for the parabola, in order to obtain the true anomaly corres- ponding to the same time, in an extremely excentric ellipse. - . - No. 23 Theory of hyperbolic motion. ....-- No. 2t Determination of the ratio of the masses of the planets accompanied by satellites, to that of the sun. ......-- No. 25 CHAP. IV. Determination of the elements of elliptic motion. - 167 Formulae which furnish these elements, when the circumstances of the primitive motion are known. Expression for the velocity, independent of the excentricity of the orbit, b2 VI CONTENTS. In the parabola, the velocity is reciprocally proportional to the square root of the radius vector ......... No. 26 Investigation of the relation which exists between the major axis of the orbit, the chord of the arc described, the time employed to describe it, and the sum of the extreme radii vectores. ........ No. 27 The most advantageous means of determining by observations, the elements of the orbits of comets. ........ No. 28 Formula; for determining by means of any number of contiguous observations, the geo- centric longitude and latitude of a comet at a given instant, and also their first and second differences. ....... No. 29 General method for deducing from the differential equations of the motion of a system of bodies, the elements of the orbits ; the apparent longitudes and latitudes, and also their first and second differences being supposed to be known for a given instant. No. 30 Application of this method to the motion of comets, supposing them to be actuated by the sole attraction of the sun : it gives, by the solution of an equation of the seventh degree, the distance of the comet from the earth. The sole inspection of three consecutive and contiguous observations, enables us to ascertain whether the comet is nearer or farther than the earth, from the sun. ..... No. 31 Method for obtaining as accurately as we please, and by means of three observations only, tlie geocentric longitude and latitude of a comet, and also their first and second differences divided by corresponding powers of the element of the time. - No. 32 Determination of the elements of the orbit of a comet, when for any instant whatever, its distance from the earth, and the first differential of this distance, divided by the element of the time is given. Simple method of taking into account the excentricity of the earth's orbit. ........ No. 33 When the orbit is parabolic, the axis major is infinite ; this condition furnishes a new equation of the sixtli degree, for determining the distance of the comet from the earth. - ........ No. 34 Hence results a variety of methods for computing parabolic orbits. Investigation of that, from which we ought to expect the greatest accuracy in the results, and the greatest simplicity in the computation. ..... No. 35 and 36 This method consists of two parts ; in the first, the perihelion distance of the comet, and the instant of its passage through the perihelion, are determined in an approximate man- ner ; in the second, a method is given of correcting these two elements by means of three observations made at a considerable distance from each other, and from them all the others are deduced. ....... No. 37 Rigorous determination of the orbit, when the comet is observed in its two nodes. No. 38 Method of determining tlie ellipticity of the orbit, in the case of a very excentric ellipse. No. 39 CONTENTS. Vll CHAP. V. General methods Jbr determining by successive approximations, the ntoiions of the heavenly bodies. - - - - - . • 23'1 Investigations of the changes which the integrals of the differential equations ought to undergo, in order to obtain those of the same equations, increased by certain terms. No. 40 Hence we deduce a simple method of obtaining rigorous integrals of differential linear equations, when we know how to integrate these same equations deprived of their last terms. -.-. ..... No. 41 An easy method is likewise deduced of obtaining continually approaching integrals, of the differential equations. .--.... No. 42 Method of making the arches of circles which are introduced into the approaching inte- grals to disappear, when they ought not to occur in the accurate integral. No. 43 Method of approximation, founded on the variation of the arbitrary constants. No. 44 CHAP. VI. Second approximation of the Celestial Motions, or theory of their per- turbations. ........ 259 Formulae of the motion in longitude and latitude, and of the radius vector in the disturbed orbit. An extremely simple form, under which they appear when we only take into account the first power of the disturbing forces. ... No. 46 Method of obtaining the perturbations, in series arranged according to the powers and products of the excentricities and inclinations of the orbits. - - No. 47 Development in series, of the function of the mutual distances of the bodies of the system, on which the perturbations depend. Application of the calculus of finite differences in this development. Reflections on this series.* .... No. 48 Formulae for computing its several terms. , - - - . No. 49 General expressions for the perturbations of the motion in longitude and latitude, and of the radius vector, the approximation being carried as far as quantities of the order of the excentricities and inclinations. , . . '- No. 50 and 51 Recapitulation of these different results, and considerations on ulterior approximations. No. 52 CHAP. VII. 0/ the secular inequalities of the Celestial Motions. • 307 These inequalities arise from the terms which, in the expression of the perturbations, contain the time, without periodic signs. Differential equations of the elements of el- liptic motion, which makes these terms to disappear. ... No. 53 If the first power of the disturbing force be solely considered, the mean motions of the planets are uniform, and the major axes of their orbits are constant. - No. 54 VIU CONTENTS. Development of the differential equations relative to the excentricicities and position of the perihelia, in any system whatever of orbits having a small excentricity and small inclination to each other. ...... No. 55 Integration of these equations, and determination by observations, of the arbitraries of their integrals. ........ Nq. 56 The system of the orbits of the planets and satellites is stable with respect to the excen- tricities, that is to say, those excentricities remain always very small, and the system only oscillates about a mean state of ellipticity, from which it deviates very little. No. 57 Differential expressions of the secular variations of the excentricity and position of the pe- rihelion. ........ No. 08 Integration of the differential equations relative to the nodes and inclinations of the orbits. In the motion of a system of bodies very little inclined to each other, their mutual in- clinations remain always very small. ..... No. 59 Differential equations of the secular variations of the nodes and inclinations of the orbits ; 1st, with respect to a fixed plane; 2dly, with respect to the moveable orbit of one of the bodies of the system. ...... No. 60 General relations between the elliptic elements of a system of bodies, whatever may be their excentricities and respective inclinations. . , - . No. 61 Investigation of the invariable plane, or that on which the sum of the masses of the bo- dies of the system, multiplied respectively by the projections of the areas described by their radii vectores in a given time, is a maximum. Determination of the motion of two bodies, inclined to each other at any angle whatever, - . - No. 62 CHAP. VIII. Second method of approAmation of the celestial motions. - 352^ This method is founded on the variations which the elements of the motion supposed to be elliptic, experience in virtue of the secular and periodic inequalities. General me- thod for determining these variations. The finite equations of elliptic motion and their first differentials, are the same in the variable and invariable ellipse. - No. 63 Expressions of the elements of elliptic motion, in the disturbed orbit, whatever may be its excentricity and inclination to the planes of the orbits of the disturbing masses. No. 6+ Development of these expressions, in the case of orbits having a small excentricity and inconsiderable inclination to each other. First, with respect to the mean motions and the major axes ; it is proved that if the squares and products of the disturbing forces be neglected, these two elements are only subject to periodic inequalities, depending on the configuration of the bodies of the system. If the mean motions of the two planets are very neariy commensurable, there may result in their mean longitude two consi- derable inequalities, affected with contrary signs, and inversely as the products of the masses of the bodies into the square roots of the major axes of their orbits. It is CONTENTS. IX from such inequalities that the acceleration of the motion of Jupiter, and retardation of that of Saturn arise. Expressions of these inequalities, and of those which the same relation between the mean motions may render sensible, in (he terms which depend on the second power of the disturbing masses. .... No. 65 Examination of the case, in which the most sensible inequalities of mean motion occur among the terras, which are of the order of the squares of the disturbing masses ; this remarkable circumstance obtains in the system of the satellites of Jupiter, and we deduce from it the two following theorems: The mean motion of the Jirst satellite, minus three times that of the second, plus twice that of the third, is accurately and constantly equal to zero. The mean longitude of the Jirst satellite, minus three times that of the second, plus twice that of the third, is constantly equal to two right angles. These tlieorems subsist notwithstanding any change which the mean motions of the sa- tellites may undergo, "either from a cause similar to what alters the mean motion of the moon, or from the resistance of a very rare medium. These theorems give rise to an arbitrary inequality, which only differs for eacli of the three satellites by the magni- tude of its coefficient, and which according to observations is insensible. - No. 66 Differential equations which determine the variations of the excentricities and perihelias. No. 67 Development of these equations. The values of these elements are composed of two parts, the one depending on the mutual configuration of the bodies of the system, which contains the periodic variations ; the other independent of this configuration, con- taining the secular variations. This second part is furnished by the differential equa- tions which we have previously considered. .... No. 63 Simple method of obtaining the variations which result from the nearly commensurable rela- tions between the excentricities and perihelias of the orbits ; they are connected with those of mean motion which correspond to them. They may produce in the secular expres- sions of the excentricities and of the longitude of the perihelia, terms extremely sensible, depending on the squares and products of the disturbing masses. Determination of these terms. ......-- No. 69 Of the variations of the nodes and of the inclinations of the orbits. Equations which determine their secular and periodic values. - - - - No. 70 A simple method of obtaining the inequalities which result in these elements, from the nearly commensurable relation which exists between the mean motions; they are connected with the analogous inequalities of mean motion. - - No. 71 Investigation of the variation which the longitude of the epoch experiences. It is on this variation that the secular equation of the moon depends. - - - No 72 Reflexions on the advantages which the preceding method, founded on the variations of the parameters of the orbits, present in several circumstances : method of inferring from them the variations of the longitude, of the latitude, and of the radius vector. No. 73 ERRATA Page Line 20, 3,./o/- This readThe. 28, 7, for (2"+z'}-, read {z"—:^)'-. 34, 6, /or mm, read mm'. 50, 19, Jbr from the M, read from M. 51, 12, /or their, jeac? its. 52, 16, Jbr z — z, read z — z'. 62, 11, for its, read these. 68, 5>forr.[-^),readr.f^—y 81, last line, for the second |, read |. 96, 8, ./or supply, read solely. 96, 19, /o,iL,,,,rf4l', dx dx 103, l.^or e, read c. 143' 17, ./or COS. en, rearf cos. tji<, 152, 19, Jbr u^, read v. 163, 3, /or tan. '^c, reafif tan. ^«. 166, 17, Jor value, read ratio. 174, 10, ybr COS. £. cos. £, rcarf cos. S. cos. S'. 174, ll.^re, read c. 174, 20, Jor sin «. sin. «'-, read sin. u. sin. u'. 216, 1,/orJS", readJS"'. 2 2 219, 1, /"or — , read . r r 224, 20, /or t/— T', read U'—V. 240, 2, ybr the second aQ, read a Q'. 244, 11, ybr these, read the. 256, 4,/or— 0, read=0. 266, l,JordR, read dR. 271, i;Jordf, readdt\ 284, 3, for a\ read »\ ERRATA. Page Line " a , a 285, 11, for a ■=. — -, read a = —-• a a 287, 13, for -__i(-_), read -. ef. 287, 16, dele — before — - . a - 299, 2, /or n + t, read nt+i. 300, 7, for e. COS. it', read e' cos. •a'. 306, 5, /or m', rearf to. 315, l,fx"dr, readfx."dR. 318, 5, yor motion, rfac/ motions. 323, \^, for m' .^^a, read m'.'/a' . 328, 20, ybr the second € J — €, readZt—Q^. 380. 2, /or in «, read (a). A TREATISE ON CELESTIAL MECHANICS, PART I.— BOOK 11. OF THE LAW OF UNIVERSAL GRAVITATION, AND OF THE MOTIONS OF THE CENTRES OF GRAVITY OF THE HEAVENLY BODIES. CHAPTER I. Of the law of universal gravitation, deduced f-om the phenomena. 1. After having developed the laws of motion, we proceed to deduce from these laws, and from those of the celestial motions, which have been given in detail in the work entitled the Exposition of the Sys- tem of the World, the general law of these motions. Of all the pheno- mena, that which seems most proper, to discover it, is the elliptic motion of the planets and of the comets round the sun, let us therefore consider what this law furnishes us with on the subject. For this purpose, let PART. I. — BOOK II. * B 2 CELESTIAL MECHANICS, X and 1/ represent the rectangular coordinates of a planet, in the plane of its orbit, their origin being at the centre of the sun ; moreover, let P and Q represent the forces with which the planet is actuated in its relative motion round the sun, parallel to the axes of ^ and of j/, these forces being supposed to tend towards the origin of the coordinates ; tinally, let dt represent the element of the time which is supposed to be constant; by the second chapter of the first bool^,* we shall have d''v . 0 = ^ + Q. (.) If we add the first of these equations multiplied by — i/, to the se- cond multiplied by x, the following equation will be obtained : ,^ d. (xdy—ydx) , „ „ 0 = — ^ ^^if + xQ—ijP. It is evident that xdy — ydx is equal to twice the area which the ra- dius vector of the planet describes about the sun during the instant dt; by the first law of Kepler this area is proportional to the time, conse- quently we have xdy — ydx = cdt, c being a constant quantity ; hence it appears, that the differential ot the firsi member of this equation is equal to cypher, which gives xQ—yP = 0, * These laws refer strictly to the motion of the centre of gravity of each planet ; it is therefore the motion of this point which is determined, and by the position and velocity PART I.— BOOK II. 3 it follows from this, that the forces P and Q are to each other in the ratio of cT to ^ ; and consequently their resultant must pass through the origin of the coordinates, that is, through the centre of the sun, and as the curve which the planet describes is* concave towards the sun, it is evident that the force which acts on it, must tend towards this star. The law of the areas, proportional to the times employed in theic description, leads us therefore to this first remarkable result, namely, that the force which solicits the planets and comets, is directed towards the centre of the sun. 2. Let us in the next place, determine the law according to which this force acts at different distances from this star. It is evident that as the planets and the comets alternately approach to and recede from the sun, during each revolution, the nature of the elliptic motion ought to conduct us to this law. For this purpose, let the differential equations (l) and (2) of the preceding number be resumed. If we add the first, multiplied by dx, to the second, multiplied by dy, we shall obtain dx.d''x + dy.d''u , „ , ^ , 0= -—^ — ^+Pdx -!- Qdy ; which gives by integrating of a planet, we always understand, unless the contrary be specified, the position and ve- locity of its centre of gravity ; hence it is evident, that the equations of the motion of a material point, which have been given in the second chapter, are applicable in the present case. * The areas being proportional to the times, the curve described is one of single curvature, {see Book I. page 28, Notes), therefore two coordinates [x, y) are sufficient to determine the circumstances of the planet's motion. As the curve described by the planet is con- cave to the sun, it is plain that in the equation —pr= P; -jj- must be taken nega- tively, because the force tends to diminish the coordinates. See Book I. Chapter II. page 31. 4 CELESTIAL MECHANICS, 0= -^^^^ -V2J\Pdx H- Qdy\* the arbitrary constant being indicated by the sign of integration. oodii^^^ u fix Substituting instead of dt, its value — - — - — , which is given by the lavs^ of the proportionality of the areas to the time, we shall have For greater simplicity, let us transform the coordinates x and j/, into a radius vector, and a traversed angle, conformably to the practice of astronomers. Let r represent a radius drawn from the centre of the sun to that of the planet, or its radius vector ; and let v be the angle which it makes with the axis of x, we shall have then, xz=.r. cos. v; y =.r. sin. r ; r ■=. y/a* + ij* ;t from which may be obtained, ■dx''-\-dy''-=.r''.dv^-\-d7''' ; xdy — ydx zz r'dv. If the principal force which acts on the planet be denoted by (p, we shall have by means of the preceding number, . P z= (p. COS. t; ; Q = (p. sin. i; ; 9 =:\/P*i-Q* ; which gives Pdx+Qdyz=.(pdr ; dx -4- dii' * The equation 0 = ^^ h '^■/{Pdx + Q(/y), has been already deduced in No 8 ; by substituting for dx^ and dif- their values in terms of the polar coordinates, we obtain — — p -J — — — [- 1J <p.dr = 0 ; hence if <p be given in terms of r we shall immedi- diately obtain the velocity at any distance from the centre of force. f The most obvious way of determining the position of any body, is by means of rectan- gular coordinates, in which case the differential equations of motion are symmetrical ; however, as the polar coordinates involve directly the quantities which are required to be known in astronomical investigations, namely, the distance, longitude and latitude of a planet, astronomers make use of these coordinates in determining the circumstances of its motion, &c. PART L— BOOK II. 5 and by substitution we shall have * dx = dr. COS. D — (/u. sin. v. r, dy — dr. sin. v + </t;. cos. v. r, •/ (/j:* + d\)' = f/i-. (COS. 'v+ sin. ';;) — 2rfr. dm. sin. v. cos. u + 2rfr. rfur. sin. u. cos.r + dv^. r^. (sin. "v + COS. 'i;)= t?r^ + dv°. r^; xdy = r. cos. u.(c?r. sin. v + rdv. cos. u) = rdr. sin. f. cos,u + rfu.r*. COS. *r, ydx=r. sin. u. (rfr. cos. « — r. rfu. sin. v)=rdr. sin. v. cos. u — rfur'. sin. "v, •/ xrfj/ — ydx = r". dv ; Pdx = (f. cos. r. (dr. cos. t;— rc/i). sin. u) ; Q,dyz=<(>. sin. u.(rfr. sin. i>+ rdv.coi.v), V Pix+Qrfy = ?irfr.(cos.^u+sin. ^d), +(prft). (r.cos. r. sin. «— r. cos. v. sin.ii) (T (dx'' + du'') = <t>dr; therefore by substituting in the equation — ^ — —rr:: + 2f(Pdx+ Qdy) = 0, [xdy—ydxy we obtain ijl^lj^+Ii ^ g /©t/r = 0 ; and • • l—ch"—T\ 2f^dr). dv" = c^dr" ; as the r «D^ ■ variables dv and (/)• are separated in the equation dv = . j " can r.V—c"—2r\fq)dr be integrated and constructed, the radical ought to be affected with the sign ±, when I) and r increase the same time, the sign is +, and in the contrary case the sign is — ; these circumstances depend on the initial impulse of the planet. The determination of v, or of the orbit described by a body, when the law of the force (p is given, is called the in- verse problem of central forces, the expression for dv coincides with that given by Newton in Prop, il, Lib. 1st. Princip. for it is there demonstrated that XY. XC = O T^ 0\'^ XY ' , from the construction it is evident that -rrrrr = dv, that IN = dr. Q A X. XC that Q;=c, and finally that A = r, and as Z« OC -^ , and ABTD = the square ot XY ^- ^'^ the velocity, V ABTD — Z- = v^ —fifdr— f_': -^^ = dv = ^V.^yiVT) — Z= cdr ■■ -H by r. r\i/—c'—'lr"f(pdr If the force <p be as any power w of the distance, then 2/<pdr= 2/r"dr (= the square of the velocity) =i'4- -- . Z'"^' — . a""*"^ (a being the initial distance), hence ■" ^» + l n+l cdr dv = V_ c'— bh-- -I — r"+3 + — ^ " " " \ as i is tiie velocity of projec- w4-l n + l 6 CELESTIAL MECHANICS, from which we may obtain, dv=: , ^gl^...^. (3) r.y/ — c* — 9.r'J'<pdr This equation will give by the method of quadratures, the value of V in terms ofr, when (pis a known function of r, but if, this force being unknown, the nature of the curve which it makes the planet describe, be given, then, by differentiating the preceding expression of 9.f(pdr, we shall have, to determine <p, the equation tion, if p be the peqiendicular on the tangent at this point, c QC p b, and b"=.m- a"+i, •• dv= — ' — , at tlie apsides »■• /_(p2 + /-),„2an+l__:i_. yn+3 ^ _1_ „b4-2 ,S _2_ p=a, dr = 0, and •/ r= ^—r'=- , . , lience _ Ir + ■ .(a""*"' — >■"+*) — pl> = 0, by squaring this equation, we get b-r' 2 2 J . o»+i r- . r"'*"3 — p"b- = 0. When n is even, this equation may have four possible roots, when it is odd, it can only have three ; but as this equation is the square of the given equation, some of the roots are in- troduced by the operation, so that the equation to the apsides can never have more than two possible roots, consequently no orbit can have more than two apsides, i. e. there are only two different distances of the apsides, but there is no limit to the number of repetitions of these, without again falling on the same points, if ?2 = — 3 or a greater negative number, the equation can have only one possible root, and the orbit but one apsid. If in the equation — j- + -r-7-" +2/?'£?'")— be substituted in place of r, it becomes c2_ /_ 1 J\ 2f(p. —T! which is a much more convenient form, particularly when the PART I.— BOOK II. dr The orbits of the planets are ellipses, having the centre of the sua in one of the foci ; if, in the ellipse, is- represents the angle which the axis major makes with the axis of x, moreover if a re- presents the semiaxis major, and e the ratio of the excentricity to the semiaxis major, we shall have, the origin of the coordinates being in the focus, 1 +e. cos. {y — ut) ' which equation becomes that of a parabola, when t? = 1, and a is in- finite, it appertains to an hyperbola, when e is greater than unity. law of the force being given, the nature of the orbit is required; for instance the equation in page 2rf2, page 5 becomes, when — is substituted for r then differentiated, and the result divided by W^"^ ■ A f ,> , 9 /'^''^ oX , 1 1-4-e. cos. (u— sr) j.^ . . _ . d^z e.cos.(u-ro) d-z „ 1 cV differentiating twice -— = ~ — -i, '.--rir -H- = -75 Tx' "•■ ^ = rf„8 - «.(!— e«) ' • dir ^~ ~ «.(1— e»)' * ^ ~ rt.(l— e^) c'".(r*.(/'D*4.rfr''') c' c'.dr" * — ' ■ ■ = — ^ — ^-— r= — Ifiidr, • • by differentiating and dividing by r*.dv r- r^.dv by dr we obtain d.l—--A—(p. r \r*dv^/ t The greatest and least values of r correspond to v—vs^ijr, u— ot'=0, •.• they are re- spectively 0.(1 + e), «.(! — e), consequently they lie in directum; hence it is easy to per- ceive, that when <p. varies as — , the apsides are 180° distant, and vice versa. 8 CELESTIAL MECHANICS, This equation gives dr'' 2 11 and consequently c* 1 « = a.(\—e^y r X ) therefore, the orbits of the planets and comets being conic sections, the force (p is reciprocally proportional to the square of the distance of the centres of these stars from that of the «un. Moreover we may perceive, that, if the force (p be inversely as the square of the distance, or expressed by —^ , h being a constant co- efficient, the preceding equation of conic sections, will satisfy the dif- ferential equation (4) between r and v, which gives the expression of if, h c* when (p is changed into — j- . We have then h = — — ^, which 1 _ I-t-e. cos.(u — ig) dr- _/e.sm.{v — ro) \^ a.(l — e") _ I (2.(1— e= \ ! 2n.(l — eM „ (v — 0-) •.• I -^— — I — -J- 1 ■=. e'-, COS. (v — -of ■=. e' — t'. sin. cos, dr- 12 1 1 °(ti — a-), •.• -TT^ ',-\ Ti T X ;; r > and the ditferential of the se- ^ r\dv' r^ a.(l—e- r a^.{l — e*) 2 2 1 cond member divided by dr will be equal to -^ — , consequently we have r' a.(l — er)' r- ' the value of dr- \ c- c- c- 1 c' c- , f dr- \ dr PART I.— BOOK II. 9 forms an equation of condition between the two arbitrary quantities a and e, of the equation of a conic section ; therefore the three arbitrary quantities a, c, and ra-, of this equation, are reduced two distinct quantities, and as tlie differential equation between r and v, is only of the second order, the finite equation of conic sections is its complete integral.* From what precedes, it follows, that, if the curve described is a conic section, the force is in the inverse ratio of the square of the distance, and conversely, if the force be inversely as the square of the distance, the curve described is a conic section. S. The intensity of thet force ?, with respect to each planet and c* comet depends on the coefficient — r- .- : the laws of Kepler fur- ^ a{l — <?") ^ nish us with the means of determining it. In fact, if we denote the time of the revolution of a planet by T; the area, which its radius vector describes during this time, being the surface of the planetary ellipse, it PAET I. BOOK II. c * Conversely, when <p = — , the preceding equation of conic sections will satisfy the differential equation (i) between r and v, and h becomes = ■ -. , '■• the three c.(l — e^j arbitrary quantities are reduced to two distinct ones, and this is the required number of arbitrary quantities, for the differential equation between r and v being of the second order, the number of arbitrary quantities introduced by the double integration is two, so that the equation of conic sections is the complete integral of this differential equation. ■)- The two first laws of Kepler, are sufficient to determine the ratio which exists be- tween the intensities of the action of the sun on each planet, at different distances of the planet from the sun ; by means of the third law we are enabled to find the relations which exist between the respective actions of the sun on different planets. As — — — , which expresses the intensity of the force for each planet, at the unity of its distance from the sun, depends on the tliree quantities a, e, c, which have particular values for each planet, we cannot determine without the third law, whether it changes, or remains the same, in passing from one planet to another. 10 CELESTIAL MECHANICS, will be 7r.a*.v 1 — e%* tt being the ratio of the semicircumference to the radius ; but, by what precedes, the area described during the instant dt, is equal to i.cdt; therefore the law of the proportionality of the areas to the times of describing them, will give the following proportion : i.cdt : ira\\/l— e* :: dt : T: •onsequently 27r.rt^^/l— e* c= J. . With respect to the planets, the law of Kepler, according to which the squares of the times of their revolutions, are as the eubes of the greater axes of their ellipses, gives T' = k'.a^, k being the same for all the planets ; therefore, we have c = 2^Vg^(l^^^ k 2fl.(l— e') is the parameter of the orbit, and in different orbits, the values of c are proportional to the areas, described by the radii vectores in equal times ; therefore these areas are as the square roots of the pa- rameters of the orbits. This proportion obtains also, for the orbits described by the comets, compared either among themselves, or with the orbits of the planets ; this is one of the fundamental points of their theory, which corresponds so exactly to all their observed motions. The greater axes of their orbits, and the times of their revolutions, being unknown, we compute the motion of these stars, on the hypothesis that it is performed in a • The area of the ellipse being equal to that of a circle, whose radius is a mean propor- tional between the semiaxes a and av'l— e* ; it must be equal to iraK^l—e'. PART I.— BOOK II. 11 parabolic orbit, and expressing their perihelion distance by D,* we suppose c = — ^^^ , which is equivalent to making e equal to unity, and a infinite, in the preceding expression of c ; consequently, we have relatively to the comets, T' = k^.a^, so that we can determine the greater axes of tlieir orbits, when the periods of their revolution are known. The expression for c gives. C* 49r* fl.(l— £') 7.« > therefore we have c 2 * The polar equation of the parabola is r = ~ -; •.• when v — v = 0, i. e. •^ ^ r+cos. (u— cr) at the perihelium, r =. — ' =— =D, :• a(l — e»)=2D. Nowthis is the same thing, (J g2\ as if a was made infinite, and e= to unity, in the equation, rz=:a.- -, which expresses the distance of the nearest apsis from the focus of the ellipse, for substituting for the ex- centricity its value V^a^—A^, r becomes equal to a.{— filZ—l | — as (i* — af) ~ — -2.' and as v'a* — apz=a — +( }• — = when a is infinite 2a '^ 2 o a.{a—a +-^) „ £_ f — ''^ — JL and it is evident that e is equal in this case to 2 ' ~ 2a * unity. •.• If we suppose that the synchronous areas are as the square roots of the parame- ters, or c = , we will have ; . dt : ^tat ^/2D :: dtiT; :' 1 —/c* a'- k 2k \ The constant ratio which c bears to the square root of 2D, is that of 2x '.h, which is the same for all the planets; -^, or --p^-rT- is the value of the "^ «' 0.(1 — e'; 12 CELESTIAL MECHANICS, The coefficient , , being the same for all the planets and comets, it Ic' force ip at the unity of the distance of a planet from the sun. The accelerating force of the planets being the same at equal distances from the sun, it follows that the moving force will be proportional to the mass; and if all the planets descended at (he same instant, and without any initial velocities from different points of the same spheric surface, of which the centre coincided with that of the sun, they would arrive at the surface of the sun, being tupposed spheric, in the same time ; here, we may perceive, a remarkable analogy between this force and the terrestrial gravity, which also impresses the same motion, on all bodies situated at equal distances fiom its centre. If the apparent diameter of the sun be observed accurately vvith a micrometer, it will be found to vary in the subduplicate ratio of his angular velocity ; from this phenomenon the equable description of areas may be inferred ; for as the apparent diameters of the sun are inversely as the distance of the sun from the earth, the angular velocity of the sun must be inversely as the square of the distance of the sun from the earth, therefore the product of the diurnal motion into the square of the distance, i. e. the small area must be constant. If the sun's mean apparent diameter be called m, and his least apparent diameter m — n, his appa- rent diameter at any other time, will be m — n cos. z, z being the angular distance of the sun from the point where his diameter is least, lience it may be inferred, that the orbit is ellip- tic ; for as the distance is inversely as the apparent diameter, r:=i — -, when m — n cos. (i' — ■sr) r is greatest, v — ■az^O, when least v — CT=:jr, •.• viu- — nr cos. (u — -z) = j('« — n), x being the greatest distance, and mr = s (m — n) -\- nr. (cos. v. — «r), let (m — ?;). x = nx', and then 7nr = ?i(r. cos. (v — to) -|- x), :• m : « : : r. cos. (v — t!!)-\-x' : r ; now r. (cos. [v — w) is equal to a part of the axis intercepted between a perpendicular let fall from the sun's place on this axis, and the place the earth is supposed to occupy, and x' is a constant quan- tity, •.• producing the axis in an opposite direction from the sun, till the distance from the earth is equal to x', and erecting a perpendicular to the produced axis at the extremity of its production, x -\- r cos. [v — ■cr) is e(iual to the distance of the sun from this perpendicu- lar, and as it is to r the distance of the sun from the earth, in a-given ratio of major ineijua- lity, namely m : n, it follows that the curve is an ellipse of which the directrix is a perpendicu- lar, erected at the extremity of x'. This conclusion might also have been inferred fi-om th« polar equation to the ellipse r = ~^ = a(l— e'). (1-f ecos. (u— w))-'. '^ H-ecos. (y — ot) Kepler directed his observations to the planet of Mars, of which the motion appeared te be more irregular, than the motion of the other planets, and by determinmg several di^ tances of the planet from the sun, and tracing the orbit which passes through them all, it will appear that this orbit must be an ellipse, of which the sun occupies one of tlie foci, it PART I.— BOOK II. IS follows that for each of these bodies, the force ip, is inversely as the square of the distance from the centre of the sun, and that it only va- ries from one planet to another, in consequence of the change of dis- tance ; from which it follows that it is the same for all these bodies sup- posed at equal distances from the sun. We are thus conducted, by the beautiful laws of Kepler, to consider the centre of the sun as the focus of an attractive force, which, decreasing in the ratio of the square of the distance, extends indefinitely in every di- rection. The law of the proportionality of the areas to the times of their description, indicates that the principal force which solicits the planets and comets, is constantly directed towards the centre of the sun ; the ellipti- city of the planetary orbits, and the motions of the comets which are per- formed in orbits, which are very nearly parabolic, prove, that for each planet and for each comet, this force is in the inverse ratio of the square of the distance of these stars from the sun ; finally, from the law of the squares of the periodic times proportional, to the cubes of the greater axes of their orbits, i. e. from the proportionality of the areas traced in equal times by the radii vectores in ditlerent orbits, to the square roots of the parameters of these orbits, which law involves the preceding, and is applicable to comets ; it follows, that this force is the same for all the planets and comets, placed at equal distances from the sun, so that in this case, these bodies would fall towards the sun, with equal velocities. 4. If from the planets we pass to the consideration of the satellites.! we find that the laws of Kepler being very nearly observed in their mo- tions about their respective primary planets, they must gravitate towards the centres of these planets, in the inverse ratio of the squares of their distances from these centres ; they must in like manner gravitate very nearly as their primaries towards the sun, in order that their relative mo- tions about their respective primary planets, may be very nearly the same can also be sliewn that the angular velocities are inversely as the squares of the distances from the sun, from which it fblluvvs that the areas are proportional to the times. 14 ^ CELESTIAL MECHANICS, as if these planets were at rest. Therefore the satellites are solicited to- wards their primaries and towards the sun, by forces which are inversely as the squares of the distances. The elliplicity of the orbits of the three* first satellites of Jupiter is inconsiderable ; but the ellipticity of the fourth satellite is very perceptible. From the great distance of Saturn we have not been able hitherto to recognise the ellipticity of the orbits of his satellites, with the exception of the sixth, of which the orbit appears to be sensibly elliptic. But the law of the gravitation of the satellites of Jupiter, Saturn, and Uranus is principally conspicuous in the rela- tion which exists between their mean motions, and their mean dis- tances from the ceiitre of these planets. This relation consists in this, that for each system of satellites, the squares of the times of their revo- lutions are as the cubes of their mean distances from the centre of the " planet. Therefore let us suppose that a satellite describes a circular orbit, of which the radius a is equal to its mean distance from the centre of the primary, T expressing the number of seconds contained in the duration of a sidereal revolution, and tt expressing as before the ratio of the semiperiphery to the radius, — '—— will be the small arc described by the satellite in a second of time. If, the attractive force of the pk' * The frequent recurrence of the eclipses of the satellites, enables us to determine the synodic revolution with great accuracy : and by means of this revolution, and of the motion of Jupiter, we can obtain the periodic time. The hypothesis of the orbits being very nearly circular, in the case of the first and second satellites, is confirmed by the pheno- mena, for the greatest elongations are always very nearly the same ; besides the supposition of the uniformity of the motions, satisfies very nearly the computations of the eclipses. The distances of the satellites from the centre of Jupiter, may be found, by measuring with a micrometer, their distances from this centre, at the time of their greatest elongation, and also the diameter of Jupiter at this time, by means of which, these distances may be obtained in terms of the diameter; however they cannot be determined with the same preci- sion as the periods of the satellites. As it is necessary in a comparison of a great nu nber of obsenations, to modify the laws of circular motion, in the case of the third and fourth •atellites, but especially in the case of the fourth, we conclude that the orbits of these sa- tellites are elliptical. PART I.— BOOK II. 15 net ceasing, the satellite was no longer retained in its orbit, it would recede from the centre of the planet along the tangent, by a quantity equal to the versed sine of the arc , that is by the quantity* ■ ; therefore this attractive force makes it to descend by this quantity, to- wards the primary. Relatively to another satellite, of which the mean distance from the centre of the primary is represented by «', 7" being equal to the duration of a sidereal revolution, reduced into seconds, the descent in a second will be equal to , ■; but if we name (p, (p', the attractive forces of the planet at the distances a and a', it is mani- fest, that they are proportional to the quantities by which they make the two satellites to descend towards their primary in a second ; therefore we have 0:0 •• — = — : — — — . The law of the squares of the times of the revolutions, proportional to the cubes of the mean distances of the satellites from the centre of their primary, gives v T* : r' :: a' : d* : From these two proportions, it is easy to infer 1 1 <?:?>:: a* d' consequently, the forces 9 and 9' are inversely as the squares of the dis- tances a and d. • T: 1" :: 2ax : arc described in a second, on the hypothesis that the motion is uni. form, the versed sine of this arc = ^~.. As the orbits of all the satellites are notd- 2a 1^ liptic, we cannot determine from the nature of the orbits, whether the force for each satel- lite in particular, varies inversely as the square of the distance or not. l6 CELESTIAL MECHANICS, 5. The earth having but one satellite, the ellipticity of the lunar orbit is the only phenomenon, which can indicate to us the law of its attractive force ; but the elliptic motion of the moon, being very sen- sibly deranged by the* perturbating forces, some doubts may exist, whe- ther the law of the diminution of the attractive force of the earth, is in the inverse ratio of the square of the distance from its centre. Indeed, the analogy which exists between tliis force, and the attractive forces of the sun, of Jupiter, of Saturn, and of Uranus, leads us to think that it follows the same lawt of diminution ; but the experiments which have been instituted on terrestrial gravity, offer a direct means of verifying this law. Fort this purpose, we proceed to determine the lunar parallax, by • The orbit of the moon differs sensibly from the elliptic form, in consequence of the action of the disturbing forces, and the variation of its apparent diameter shews, that it de- viates more from the aVcuZar form, than the orbit of the sun. The first law of Kepler may be proved to be true, in the case of the moon, In the same manner as for the sun, namely, by a comparison of her apparent motion, with her apparent diameter. Indeed, if great accuracy is required, the observations ought to be made in the syzygies and in the quadra- tures ; for in the other points of the orbit, the disturbing force of the sun deranges the proportionality of the areas to the times employed in their description. See Princip. Math. Lil). 1. Prop. 66. and Lib. 3, Prop. Sand 29. f Newton demonstrates that th? force which retains the moon in her orbit, is inversely as the square of the distance, in the following manner : if the distance between the apsides was 180°, the force would be inversely as the square of the distance, as has been already pointed out. See Note to page 7- Now the apsides are observed to advance three degrees and three minutes every month, and the law of the force which would produce such an advance of the apsides, varies in- Tersely as some power of the distance, intermediate between the square and the cube, but which is nearly sixty times nearer to the square ; •.• on the hypothesis, that the progres- sion of the apsides, is produced by a deviation from tlie law of elliptical motion, the force must vary very nearly va the inverse ratio of the square of the distance; but if, as Newton demonstrates, the motion of the apsides arises from the disturbing force of the sun, it follows, aforliori, that the force must be inversely as the square of the distance. X The value of the constant part of the parallax is deduced on the hypothesis, that the force soliciting the moon, is the terrestrial gravity, diminished in the ratio of the square of PART I.— BOOK II. 17 means of experiments on the length of the penduUim which vibrates se- conds, and to compare it with observations made in the heavens. On the parallel of which the square* of the sine of the latitude is J, the space through which bodies fall by the action of gravity in a second, is, from observations on the length of the pendulum, equal to 3'°"[",65548, PART. I. BOOK II. D the distance ; and if this parallax agrees with the observed parallax corrected for the lunar inequalities, we are justified in inferring, that the diminished terrestrial gravity and the force solliciting the moon are identically the same. • Let unity represent the radius of a sphere equicapacious w ith a spheroid, its density being supposed to be the same with the mean density of this spheroid; if the greater semi- axis of the spheroid be = 1+g, and the lesser = 1 — s, we shall have for the oblong spheroid the following equation, -—.1^=—— (1 + ^).(1 — s)*, v P = 1+g — 2* neglecting the squares and products of s and §, which is permitted as the ellipticity of the spheroid is supposed to be inconsiderable, consequently we have {:^2«, ••• in an oblong spheroid, such as would be generated by a revolution about the greater axis, the ele- vation of the spheroid above the equicapacious sphere is double of the depression below this sphere ; and if r be the radius of the equicapacious sphere, a the greater, and b the lesser axis of the spheroid, we have a — r= 2r — 2b, :• r = — - — ; if the spheroid be oblate, i. e. such as would be generated by a revolution about the lesser axis, 4* i 43- —.1^ = — . (1 — «)(l + e)^, hence j=2j, i.e. the depression inthisca«eis equal to twice the elevation, •-• 2a — 2r=r—b, andr= — — — . ' ' 3 If a sphere be inscribed in a spheroid, the elevation of any point of the spheroid above the inscribed sphere, is to the greatest elevation of a spheroid above the inscribed sphere, i. e. to the difference between the radius of the equator and seniiaxis, as the square of the cosine of the angular distance A from the axis major, to the square of radius, •.• the elevation =z {a — b) cos. *a, and as the equicapacious sphere is elevated above the lesser axis, and •/ above the inscribed sphere by a quantity equal to r — b, the ele- vation of the spheroid above the ctjuicapacious sphere =(a-^b) cos. "t^ — r-{-b:=(_a — b). 2a4-b ., / —2a + 2b\ ,,,,•• <^ ,, cos. 'A — +A, ^= I, consequently when the elevation is 0, we have 2 1 cos. "-A = — , V sin. 'a = — , and a = 35°16'. This situation is also remarkable 3 3 for being the distance from the quadrature at which the addititious force of the sun, ig equal to that part of its ablatitious force, which acts in direction of the radius of the moon's orbit. 18 CELESTIAL MECHANICS, as we shall see in the third book : we select this parallel, because the attraction of the earth on the corresponding points of its surface, is very nearly, as at the distance of the moon, equal to the mass of the earth, divided by the square of its distance from its centre of gravity. Under this parallel, the gravity is less than the attraction of the earth, by f * of the centrifugal force which arises from the motion of rotation at the equator ; this force is the - th part of the force of gravity ; consequently we must augment the preceding space by its 432d part, in order to obtain the entire space which is due to the action of the earth, which on this parallel, is equal to its mass divided by the square of the terrestrial radius ;_ therefore this space will be equal to 3'"',66394. At the distance of the moon, it must be diminished in the ratio of the square of the radius of the spheroid of the earth, to the square of the distance of this star, to effect this, it is sufficient to multiply it by the square of the sine of the lunar parallax ; therefore X representing this sine under the parallel above mentioned, we shall have ■2'*.3"'%66394, for the height through which the moon ought to fall in a second, by the attraction of the earth. But we shall see in the theory of the moon, that the action of the sun diminishes its gravity towards the earth by a quantity, of which the constant part ist • The centrifligal force at the equator is to the efficient part of the centriftigal force at any parallel, as the square of radius to the square of the cosine of latitude, i. e. in this case, 2 1 as 1 to — , -.' as the centrifugal force at the equator is the-—— th part of the gravity, the force o 288 2 1 1 at the parallel in question, will be = -^ '"ooq" "^ 3 288 432 \ m being the mass of the sun, and d its distance from the moon, a the radius of th^ moon's orbit, the addititious force = — r— > and the part of the ablatitious force, which acts in the direction of the radius vector ==-— -. 3 sin. ^-sr, -a being the angular distance from quadrature, see Kev.ton, Princip. Prop. 66 ; ••• -^C — 3 sin. 'sr) is the part of tlie sun's PART I.— BOOK II. 19 equal to the th part of this gravity ; moreover, the moon, in its re- lative motion about the earth, is sollicited by a force equal to the sum of the masses* of the earth and moon, divided by the square of their mu- tual distance ; it is therefore necessary to dimipish the preceding space by its 358th part, and to increase it in the ratio of the sum the masses of the earth and moon, to the mass of the earth ; but we shall see in the fourth book, that the mass of the moon deduced from the pheno- raena of the tides, is a — - — th part of the mass of the earth ; therefore 5o,7 the space through which the moon descends towards the earth, in the interval of a second, is equal to -^. — ^ . ■r^3'"^66394. 358 58,7 Now a representing the mean radius of the lunar orbit, and T", the duration of a sidereal revolution of the moon, expressed in seconds ; d2 disturbing force acting in the direction of the radius, which is efficient at any point; (hence it appears that it vanishes when sin. *w — . — , see Note, page 17); in order 3 ma to obtain its mean quantity, multiply this expression by dtn and it becomes • : (rfrar — Srfar. sin. »ir) = -— - {dia — --rfar-f- — rfzircos. %b), and its integral = ——{vi~- 3,3. ..„ ., ma v . 2"^ •'"Z"' ^'^ ' ^^ entire circumference, z. c. when ■»=«-, Tr""5~' ■•' the mean disturbing force := , but — : ii' the force retaining the moon in its orbit : : J**"* T^ ' ' 2re the periods of the sun and moon) ••• -rj- r: — 7^^^ Ttq' "t"^ ' = .^Q-, and — ~qjr^^ ~ 'oTo'' ■•' ■" consequence of the diminution of her gravity by the action of the disturbing force, the moon is sustained at a greater distance from the earth, than it would be if the action of the sun was removed, and as the mean area de- scribed in a given time in the primitive and disturbed orbits is the same, the radius vector is increased by a 358th part, and the angular velocity is diminished by a l79th part. * The moon being considered as a point, if it revolved about the centre of the earth, in 20 CELESTIAL MECHANICS, 9/7 * ^,^ will be, as has been already observed, the versed sine of the arc which it describes during a second, and it expresses the quantity, by which the moon has descended towards the earth, in this interval. This value of a is equal to the radius of the earth, under the above mentioned parallel, divided by the. sine of x ; this radius is equal to 6369514"''; therefore we have 6369514""'^' a- . X but in order to obtain a value of a, independent of the inequalities of the moon, it is necessary to assume for its mean parallax of which the sine is x, the part of this parallax, which is independent of these inequalities, and which has been therefore termed the constant part of the parallax. Thus, tt representing the ratio of 355 to 113, and T' being =: 2732166" ; the mean space through which the moon de- scends towards the earth, will be 2.(355)16369514"" (113/..r.(2732lG6)'" the same time in which it revolves about the common centre of gravity of the earth and moon, the central force which should exist in the centre of the earth capable of effecting this, should be ::: to the sum of the masses of the earth and moon ; for a being the dis- tance of the earth from the moon, and m,rril their respective masses, the distance y at which the moon would revolve round the earth by itself, considered as quiescent, is I , see Prin. Math. Prop. 59, Book I. and T ' =: ^—=: — ; — , , hence if a be the distance, the central force =: m-\-m', ••• as the versed sine of the arc described in a second is the space through which the moon descends in consequence of the combined actions of the earth and moon, this must be diminished in the ratio of ?n : m-J-i«' to obtain the space described in consequence of the sole action of 7«. The two corrections, wliich are here applied to the space through which a heavy body would descend at the latitude 55' 16', diminished in the ratio of the square of the distance, are in the Systeme du Monde, applied to the versed sine of the arc described in a second, hence it appears that they must be affected with contrary signs. PART I.— BOOK II. 21 By equalling the two expressions, which we have found for this space, we shall have ,3_ '2.(355)-. 35S.5S,7.6369514< ^^ ~ (USy. 307. 59,7. 3,6ii39ii'-2732l66y ' from which we obtain 10536 ",2 for the constant part* of the lunar pa- rallax, under the parallel in question. This value differs very little from the constant quantity 10540,7 which Triesnecker collected from a great number of observations of eclipses, and oft occultations of the stars by the moon ; it is therefore certain that the principal force which retains the moon in its orbit, is the terrestrial gravity diminished in the ratio of the square of the distance ; thus, the law of the diminution of gravity, which in the planets attended by several satellites, is proved by a comparison of the times of their revolutions, and of their distances, is • In order to find the constant part of the parallax, we apply to the observed parallax, all the corrections which theory males known, and we may perceive from this how the theory of gravity, by indicating the forces which act on the moon, furnishes us with the means of determining the mean motion, and the nature of the inequalities which act on it. f If in a partial eclipse of the moon, the time be noted in which the two horns of the part which is not eclipsed, are observed to be in the same vertical line, it would be easy to shew that the height of the centre of the moon at this instant, will be the same as the height of the centre of the shadow ; •.• if at this instant the height of each of the horns be observed, the mean height,which will be the heightofthecentreof the shadow, will be the apparent height affected by the parallax ; but as the centre of the shadow is diametrically opposite to the centre of the sun, the true height will bo equal to the depression of the sun, which is known from the time of observation ; •.• the ditl'erence of these heights will be the parallax of the moon for the observed altitude, by means of which we can easily determine the greatest parallax; and if in a total and central eclipse, the height of the moon be observed at the instant that it is entirely immersed, and also when it Jint begins to emerge, the mean height will be the height of the centre of the shadow as it is affected by parallax. In an occultation of a fixed star, the star's parallax vanishes, and the difference of ap- parent altitudes is = to the difference of the true altitudes -|- parallax in altitude of the moon ; hence by the known formulae we can obtain the true parallax. A constant ratio exists between the horizontal parallax, and the moon's apparent diameter at the same terrestrial latitude. 22 CELESTIAL MECHANICS, demonstrated for the moon, by comparing its motion with that of pro- jectiles near the surface of the earth. It follows from this, that the ori- gin of the distances of the sun, and of the planets, ought^in the com- putation of their attractive forces, on bodies placed at their surface, or beyond it, to be fixed in the centre of gravity of these bodies ; since this has been demonstrated to be the case for the earth, of which the attrac- tive force is, as has been remarked, of the same nature with that of these stars. 6. The sun and the planets which are accompanied with satellites, are consequently endowed with an attractive force, which decreasing in- definitely, in the inverse ratio of the squares of the distances, comprehends all bodies in the sphere of its activity. Analogy would induce us to think, that a like force inheres generally in all the planets and in the comets ; but we may be assured of it directly in the following manner. It is a con- stant law of nature, that one body cannot act on another, without expe- riencing an equal and contrary reaction ; therefore the planets and comets being attracted towards the sun, they ought to attract this star according to the same law. For the same reason, the satellites attract their respec- tive primary planets ; consequently^'ais attractive force is common to the planets, to the comets, and to the satellites, and therefore we may con- sider the gravitation of the heavenly bodies, towards* each other, as a general property which belongs to all the bodies of the universe. We have seen, that it varies inversely as the square of the distance ; indeed, this ratio is given by the laws of elliptic motion, which do not rigorously obtain in the celestial motions ; but we should consider, that the simplest laws ought always to be preferred, unless observations com- pel us to abandon them ; it is natural for us to suppose, in the first in- stance, that the law of gravitation is inversely as some power of the dis. « Besides, it follows from tlie sphericity of these bodies that their molecules are united about their centres of gravity, by a force which at equal distances solicits them equally towards these points ; the existence of this force is also indicated by the perturbations which the planetary motions experience. PART L— BOOK II. 23 tance, and by computation it has been found, that the slightest differ- ence between this* power and the square, would be very perceptible in the position of the perihelia of the orbits of the planets, in which obser- tion has indicated motions hardly perceptible, and of which we shall hereafter develope the cause. In general, we shall see throughout this treatise, that the law of gravitation inversely as the square of the dis- tance, represents with the greatest precision all the observed inequalities of the motions of the heavenly bodies; this agreement, combined with the simplicity of this law, justifies us in assuming that it is rigorously the law of nature. The gravitation is proportional to the masses ; for it follows from No. 3, that the planets and comets being supposed at equal distances from the sun, and tlien remitted to their gravity towards this star, would fall through equal spaces, in the same time ; consequently their gravity will be proportional to their mass. The motions almost circular of the satellites about their primaries, demonstr;ife that they gravitate as their primaries towards the sun, in the ratio oi their masses ; the slightest difference in this respect, would be perceptible in the motions of that satellites, and observations have not indicated any inequality depending * See No. 58 of this book ; this also follows from Prop. 45, Book 1st, Prin. For if the force which is added to the force varying in the inverse ratio of the square of the distance be called X, the angular distance between the apsides = 1 SO. . = 180.(1 — X), the square of -/l+^X ^ X being neglected, and conversely if the distance between the apsides be given, wt can determine X. The force X is supposed to vary as the distance. f See Newton Princip. Prop. 6, Book 3, where it is shewn, that if the satellite gravitated more towards the sun than the primary at equal distances from the sun, in the ratio ofd:e, the distance of the centre of the sun from the centre of the orbit of the satellite, would be greater than the distance of the centre of the sun from the centre of the primary, in the ratio of V «/ : v/ e , ••• if the difference between d and e, was the thousandth part of the entire gi-avity, the distance of the centre of the orbit from the centre of the sun, would be greater than the distance of the centre of Jupiter from that of the sun, by a th part ^ ' " 2000 ^ of the entire distance. 24 CELESTIAL MECHANICS, on this cause. Therefore it appears that if the comets, the planets and satellites, were placed at equal distances from the sun, they would gravi- tate towards this star, in the ratio of their masses ; from which it follows, in consequence of the equality between action and reaction, that these stars must attract the sun, in the same ratio, and consequently their action on this star, is proportional to their* masses divided by the square of their distance from its centre. The same law obtains on the earth ; for from very exact experiments instituted by means of the pendulum, it has been ascertained, that if the resistance of the air was removed, all bodies would descend towards its cefitre viith equal velocities ; therefore bodies near the earth gravitate to- wards its centre, in the ratio of their masses, in the same manner as the planets gravitate towards the sun, and the satellites towards their pri- maries. This conformity of nature with itself on the earth, and in the immensity of the heavens, evinces in the most striking manner, that the * The mutual attraction does not affect the elliptic motion of any two bodies when their mutual action is considered, for the relative motion is not affected when a common velocity is impressed on the bodies, ••• if the motion which the sun has, and the action which it experiences on the part of the planet, be impressed in a contrary direction, on both the sun and the planet ; the sun may be regarded as immovable, and the planet will be sol- licited by a force ::' to the sum of the masses of the sun and planet, divided by the square of their mutual distance ; •/ the motion will be elliptic ; but the periodic time will be less than if the planet did not act on the sun, for the ratio of the cube of the greater axis of the orbit to the square of the periodic time, is proportional to the sum of the masses of the sun and planet; however as this ratio of the square of the time to the cube of the distance, is very nearly the same for all the planets, it follows that the masses of the planets must be comparatively much smaller than the mass of the sun, which is confirmed by an estimation of their volumes. See No. 25, and Prop. 8, Lib. 3. Frincip. Math. Tlie comparative smallness of the masses is also confirmed by the laws which Kepler was enabled to an- nounce, for tliese laws were deduced from observation, notwithstanding tlie various causes which disturb the elliptic motion ; hence appears the reason why, in the commencement of this chapter, the sun was supposed to be immoveable, and to exert its action on the planets as on so many points, which do not react on the sun, neither was the mutual action of the planets on each other taken into account ; the same simplifications were employed, when the motion of a satellite about its primary was considered. PART I.— BOOK II. 25 gravity observed here on earth, is only a particular case of a general law, which obtains throughout the universe. The attractive property of the heavenly bodies does not appertain to them solely in a mass, but is peculiar to each of their molecules. If the sun only acted on the centre of the earth, without attracting in particular each of its parts, there would be produced in the sea, oscillations much greater, and very different from those which we observe ; there- fore the gravity of the earth to the sun, is the result of the gravitations of all its molecules, which consequently attract the sun, in the ratio of their respective masses. Besides, each body on the earth gravitates towards its centre, proportionally to its mass ; it reacts therefore on the earth, and attracts it in the same ratio. If this was not the case, and if any part of the earth, however small, did not attract the other part, as it is attracted by this other part, the centre of gravity of the earth would have a motion in space, in consequence of the force of gra- vity, which is impossible. The celestial phenomena, compared with the laws of motion, conduct us therefore to this great principle of nature, namely, that all the molecules of matter mutually attract each other in the proportion of their masses, divided by the square of their distances. We may perceive already, in this universal gravitation, the cause of the perturbations, which the heavenly bodies experience ; for the planets and comets being subject to their reciprocal action, ought to deviate a little from the laws of elliptic motion, which they would accurately follow, if they only obeyed the action of the sun. The satellites in like manner deranged in their motions about their primaries, by their mutual attraction, and by that of the sun, deviate from these laws. We may perceive also, that the mole- cules of each of the heavenly bodies, united by their attraction, should constitute a mass nearly spherical, and that the result of their action at the surface of the body, should produce all the phenomena of gravitation. We see moreover, that the motion of rotation of the heavenly bodies, should slightly alter the sphericity of their figure, and flatten them at the poles, and that then, the resultant of their mutual action, not pass- PART I BOOK II. E 26 CELESTIAL MECHANICS, ing accurately through their centres of gravity, ought to produce in their axes of rotation, motions similar to those, which are indicated by ob- servation. Finally, we may perceive why the molecules of the ocean, unequally acted on by the sun and moon, ought to have an oscillatory motion, similar to the ebbing and flowing of the sea. But the deve- lopement of these different effects of universal gravitation, requires a profound analysis. In order to embrace them in all their generality, we proceed to give the differential equations of the motion of a system of bodies, subjected to their mutual attraction, and to investigate the exact integrals which may be derived from them. We will then take advantage of the facilities which the relations of the masses and distances of the heavenly bodies furnish us with, in order to obtain integrals more and more accurate, and thus to determine the celestial phenomena, with all the precision which the observations admit of. PART L— BOOK II. S? CHAPTER II. Of the differential equations of the motion of a si/stem of bodies, sub- jected to their mutual attraction. 7. LET m, m', m", &c. represent the masses of the different bodies of the system, considered as so many points ; let ^, i/, z, be the rectangu- lar coordinates of the body m ; a/, y', z', those of the body m', and corresponding expressions for the coordinates of the other bodies. The distance of m' from m being equal to v/ {a^-xy + (T/'—yy + (z'—z}\ its action on m, will be, by the law of universal gravitation, equal to n^ i^—^y+ii/'—^y-^iz'—zy If we resolve this action, parallel to the axes of a\ of y, and of z, the force parallel to the axis of ,r, and directed from the origin, will be m(y—x) * W-^y+{jj'—yy-^(,z'^zy)Y^ E 2 * The force parallel to the axis of x: -7 , .,,"*> ; •: ^— x) : mm' JTD -xi 1 / / ;; , , - — r^ , .. / be differenced with V(x— x)-+(y'— y)^+(2_2)- ; and if ^y_a,,i+(^'_^)i + (2'_j)i respect to x, and then divided by m.dx, it will become _ _j[__. nim'.(x' — x).d.T: 28 CELESTIAL MECHANICS, or ^ dx ) We shall have also, 1 /* J mm" A m '< 'v/(y'— ^)'^ + {i/'—j/y+(z"—z)^ > V dx y for the action of m" on m, resolved parallel to the axis of x, and corres- ponding expressions for the other bodies of the system. Consequently if T^m' mm'' + "^'"^'' + &c • A. representing the sum of the products of the masses m, rti, »»", &c, taken two by two, and divided by their respective distances ; — . j— 7-^f * will express the sum of the actions of the bodies rn, m", kc. on m, resolved parallel to the axis of x, and directed from the origin of « 1 f '^^ \ i^j """' 'm''\dx)~ nT X V(j'— f)'+(y— y)^+(;'— z)' "^ dx mm" . ■> vi'.{x'—x) __.. ., . -^"\ dx ^/(x"-^)^■f (y-^)^ +(z"_z) » ^ • 3 (C:r'-x)»+(y =_y)'-Ks'-«)*) ' " {x"-x) 4- jT-r, ,.,,,,, .,,,., r^TT-^ 4- &c. = the sum of the actions of the bodies m', m", »»'", &c. on m, resolved parallel to the axis of ir. PART I.— BOOK II. . 39 the coordinates. Therefore dt representing the element of the time, supposed constant ; we shall have by the principles of dynamics, ex- plained in the preceding book, O = m . — \ i . dt^ I dx S In like manner we shall have d^u ^ dx-) di* Idt/S dt^ I dz ]• 0 = m.- If we consider, in the same manner, the action of the bodies m, m", &c. on m' ; that of the bodies m, m', on m", and so of the rest, we shall have the following equations, namely, dt* \dci/y ' dt* Xdy'S ' df \dz"S The determination of the motions of m, m', m", &c., depends on the integration of these differential equations ; but as yet they have not been completely integrated, except in the case in which the system is composed of only two bodies. In other cases, we have not been able to 90 • CELESTIAL MECHANICS, obtain but a small number of perfect integrals, which we proceed to develope. 8. For this purpose, let us first consider the differential equations in •r, x\ af', &c. ; if we add them together, observing at the same time, that by the nature of the function x, we have we shall obtain, 0 = l.m. ^ • We shall have also, 0 = S,m. -— ; etc t»* d'z 0= l.m. . Let A', Y, Z represent the three coordinates of the cen- tre of gravity of the system j we shall have by the nature of this centre l..m l.m 2,m therefore we shall have d'X ^ d^Y ^ d*Z ^ = -dT' "" = -dT'' "^ = -dT' and by integrating, we shall obtain X = a+bt ; Y = a'+ b't; Z = a"+b"t ;t • Suppose that there are only three bodies, then l.m.—— =^-j-y + \ 'TTj'r \~7T' / _ m'm.{(x'—x)—(x'—x)) ^ mm"({x"—a:)—{x"—x) ) _ , -i- , — "' '" U*^ ^-^ )—v'^ ■^); 3^ _Q ti)g sgjpg proof may be extended to any num» ber of bodies. PART I.— BOOK II. SI a, a', a", b, b', b", being constant arbitrary quantities. We may per- ceive by this, that the motion of the centre of gravity of the system is rectilinear and uniform, and that consequently, it is not deranged by the reciprocal action of the bodies composing the system ; which agrees with what has been demonstrated in the fifth chapter of the first book. Resuming the differential equations of the motion of these bodies, and multiplying the differential equations in y, y', y'', Sec, respectively by a:, of, x'\ &c., and then adding them to the differential equations in *, **, *", &c. multiplied respectively by — y, — y', — y" , &c. ; we shall obtain \ r x"d'y"—y"d'x" 1 , , but from the nature of the function x, it is evident that c rfx ) ^ dx } „ X. ■m—m-''-' grating," = a, and X=at+b, the constant quantity a depends on the velocity of the centre of gravity at the commencement of the motion, and b depends on the position of this centre, at the same instant. S2 CELESTIAL MECHANICS, consequently,* by integrating the preceding equation, we shall obtain In like manner we shall have, xdz — zdx ^ f xdz — zdx > = ^•"^•1 — dF-y^ f ydz—zdy \ . 1 'dt r c" ='E.m. c, c', (f, &c. being constant arbitrary quantities. These three integrals involve the principle of the conservation of areas, which has been ex- plained in the fifth chapter of the first book. Finally, if we multiply the differential equations in x, x', x', &c., re- spectively by dx, dx, dx", &c. ; and those in y, y', y' , &c. respectively by dy, dy', dy", &c. ; those in ;:, z', z", &c., respectively by dz, dz', dz", &c. ; and then add them together, we shall obtain _ _, (dx,d'x4-di/.d''i/4-dz.d^z) , , O — ^.m.— ' ■% , — dx, T dr * Suppose that there are only three bodies, then i/( j— ) + v(-p- l'*'-^ Y"/^)"" n".( '^'^'y"—if'd'^x' \ _ mm'.{y{,x'—x)—y' ( x'—x) ) ( [x-xy+[y'—y)'-\-[^-zy ) ^ , mm"(y{i^'—x)—y"{x"—x) ) , m"m( )y'(x»—x')—y''{x'—x') ) , '^ {(x'-xy-\-(y"—yY+[^'-z)^)i + ( (x/z-y) - My"-y') ' + (^"-^r ) * mm'(x{y' — y) — x'(y' — ;/) ) ^ mm\x[y" —y) — x"{lj' — ^j)) , ( {x-xY+{i,—y)'^{z-zY-y —{(3!--xY^{y"-y) -^{J< -z^f _ »»^/m'(y(y"-y)-^'(y/-y')) ,_ t Bymultiplying |^^ |^ J + &c. by ^., cf^. rfx",&c.;{ A | , ^ |, J ,+ PART I.— BOOK II. 33 and by integrating, h being a new arbitrary quantity. This integral contains the principle of the conservation of living forces, which has been treated of in the fifth chapter of the first book. The* seven preceding integrals are the only exact integrals, which we have hitherto been aljle to obtain j when the system is composed of only two bodies, the determination of their motions is reduced to differ- ential equations of the first order, which can be integrated, as we will »ee in the sequel ; but when the system is composed of three or a greater number of bodies, we are then obliged to recur to the methods of approximation. 9. As we can only observe the relative motions of bodies ; we refer the motions of the planets and of the comets, to the centre of the sunj and the motions of the satellites, to the centre of their primaries. Therefore in order to compare the theory with observations, it is neces- sary to determine the relative motions of a system of bodies, about a body which is considered as the centre of their motions. Let M represent this last body, m, m, m", &c., being the other bo- dies, the relative motion of which about M, is required ; Let (, U and y be the rectangular coordinates of M, ^+x, n + t/, y+z, those of WJ ; l-i-x', n-t-j/', y-f r", those of m', &c. ; it is manifest that x, y, z, will be the coordinates of vi, with respect to M ; that /, y', z', will be those PART I. BOOK II. F 4c, by dy, dy', dxj' , &c. and then adding these quantities together, their aggregate is equal to the differential of a considered as a function of x, x', &c. i/, y, &c. z, «', &c., and •.• it is equal to dx. * Three of these ii\tegrals are furnished by the principle of the consei-vation of areas, three by the principle of the conservation of the n.otion of the centre of gravity, and one bj the conservation of living forces. 34 CELESTIAL MECHANICS, of m' referred to the same body, and so of the rest. Let r, r', &c., re- present the distances of m, m', &c. from the body M, so that and let us also suppose m'm Vi^v- xY + {jsJ—yy + (2'-z)* mm" -L — — + &c. This being premised, the action of m on M, resolved parallel to the axis of X, and tending from the origin, will be — j- ; that of vfi on M resolved in the same direction, will be — -7-, and so of the other bo- dies of the system. Therefore, to determine^, we will have the fol- lowing differential equation : dt' r' " and in like manner. d'n ""- di^ dt' mz PART I.— BOOK II. 39 The action o£ M on m, resolved parallel to the axis of .r, and directed from the origin, will be —, and the sum of the actions of the bodies m; m", &c. on m, resolved in the same direction, will be — . m f -7-;- j ; consequently, we will have and substituting in place of — -1 its value S.^, we will obtain di r^ dt^ r* f ' m I dx b in like manner, we will have d*z , Ms ^ mz 1 C d\ 1 ,_, F 2 * — . •{ -7— > is equal to the sum of the actions of the bodies m', m", &c. on m, re- »i (^ ax 3 solved parallel to the axis of x, •.• if we add to this expression the action of M oh m. which is equal to , we will have the actions of all bodies of the system on m, ana r' d^ip^x) Mx •■• hj the principles of dynamics established in thfr first book, — — \r ~^ 1_ (d\l _ m ' \dx $ ~~ ' sa CELESTIAL MECHANICS, If in the equations (I), (2), (3), we change successively the quantities m, X, 1/, z, into m', x', y', s! ; tw^ a!', y", z', &c. ; and reciprocally, we will obtain the equations of the motion of the bodies w, m", &c. about M. If we multiply the differential equation in ^, by M+S.m. ; that in x, by m ; that in «', by wi', and performing similar operations on the other differential equations ; by adding them together, and observing that by the nature of the function a, we have ''-m-m^^-^ we will obtain from which we obtain by integrating • The differential equation in ^, becomes by this multiplication, (M-j-2.m.) — _ M.2. — — 2.W.2. ^ = 0 ; and if the differential equations in s, jf, x", &c. be multi- plied by m, m', m", &c., respectively, and then added together, their sum will b« = if this expression be added to the preceding, we will have, observing the quantities which ^j + 2. TO, |-^ 1=0, and by integrating we have (M+2.OT.)- 1 "^ | +2.m. j -^ ^ =</, V (Af+2.«)^4- 2.m^ = c+dt, and •.• if ^^i_- -a -^^^j;^ = b, we shaU have^_ ^e expression given in the text. PART I.— BOOK II. 37 a and b being two constant arbitrary quantities. We will obtain also n=a' + b't ^-""y -; a', ft', fl", 6", being constant arbitrary quantities : we shall thus obtain the absolute motion of M in space, when the relative motions of w, m'. Sec, about it, are known. If we multiply the differential equation in x, by and the differential equation in j/, by and in like manner, the differential equation in a/, by and the differential equation in y, by Af+S.7» ' 38 CELESTIAL MECHANICS, and if the same operations be performed on the coordinates of the other bodies of the system, by adding all these equations together, and observing that by the nature of the function x, we will obtain dr M-^^.m dt' M+^.m dt' .« ax • Performing these operations, the difFerential equation in x becomes ~ — my. —r-^ — . , M mx ( d\ ] m d- x , Mmx 2 mil m mx l.mi/ ( d?. "t , ,. ■ . • -zrr: . 2. — - . l.mu — — ■ — . J — — )- ; and corresponding operations being per- M+2.m r^ ^ M+~.m \ c/x ) ' ^ ° ^ formed on tlie differential equations in x', x", &c. we obtain, by adding them all together, d^x , , i/x mx C d>^ t l.mij d^x , " dt* T^ ^ r^ ^ ^ \ dx S ^ M + 2.m dt'' ^ 2.my.M ^ m, ^.m.^-mj, ..J^ _ J=?f^ 2. j--j ; multiplying the differ. Af+2.m. >•» + M+2.m ' r» M+2.m I dx i ?.mx , , , '^■mx ential equations in y,i/',i/", &c. by mx-m. ^^_^^ ^^^ , « *' -»« -^^^s.,^ ' &c, we OB- d^u xu my 5 '^^ ? *" tain for the equation in ^, mx.—^+ M.m.^ + mx.2.— ^'I'^ij^ M+Tm.' i-y mM y 5,»nx ^ my 'S.-mx 5^? . ]f the same operation be performed for the equations in ^r'andy", &c. we obtam, by addmg ihes0 equations, and concinnating rf*y . ,, m.xij my „ \ dx \ d y 2.mx.2.^ + ,V.2.-^+2.;«r.2.-f -2.x.|_ \ ^ 2.m. ^^. ..mx.^ iW + 2.m PART I.— BOOK II. Sy of which equation the integral is Const"'. = £.ff». ^ ^ ,f — -— ^rp-- S.W. -jf at M-{ S.7» rf/ E.m?/ dx + — 7 — t: — . 2.WJ.- orc =: M^^.m ' "•'"• (f? ' M.l.m.^^^Jl:^^^ x.mm'. { i^'--Udy'-dy)-iy'-y-^d^-d^) | . ^.^ . 2.— ^+2.ni«.2. < — p- \ ; JV/+2.m this equation being added to the equation obtained, by taking the sum of the equations i* X, K, &C. gives f 'i.my. d's 2 m.x d^y "» S.tkx.S. I </a ") l.my.'S. f <^* 1 1 M+S.nt ■'"■'S« M+2.m df J Ai+2.m 1^3 ~" M+2.m \'dx ]' the quantities which destroy each other, by the opposition of signs are omitted. se* page 31. The first term of the second member of this equation is evidently an exact differential, »ee page 2, and the integral of the remaining terms which do not vanish r: —■ . 2.m. Af+2.»«- cT/ /'S.m.dy dx 2-mj; du , f 'S.nidx du rr . 2.OT.-T ■. 2.7n.-^+ / . 2.m.-2-. M-»-2.m dt M + 2.m dt ^ J M+2.m dt 40 CELESTIAL MECHANICS, * being* a constant arbitrary quantity. By a similar process we may ob- tain the two following integrals : • If there are but three bodies *"J »"^y __ m'x'.m'dy m"x". m"du" mx.m'dy' Af+m+;«'^m" * dl (3/-|»i-f.«i ■\n}:')d~ {M-\,n\m' \vi: ' )dt ~ kMA-m-\-m' ^mn^dr mx.m'dy'i m'x.mdy m'x'.m"dy" m'x".mdy (M+m+m'Jrm'-)dt (.M + ,« + m' + m") d~{M-\-m-lrin' + m").dt ~ (M-\- m + m'+")rf<~ m"x".m'di,' my.mdx m'y'.m'dx m"y'.m"dx" )<lt M+m+m -^m'.)dt '^ (M-^m + nf +m')dt'^ {M-\-m-{.m'-Jfm')dt "^ (Ai-)-m+m'+m"> , . my.m'dx' my.m"dx" m'y'.mdx m'j/.m"ds;' (M-]rm-\.ni^m')dt "'■(M+ ra-f „;'+,«//)</<+ yM^m\m'^m")dt '^ {M\m + m' + m .)dt , m"y".mdx m" y" m'dx ^\M^;;^^^;7+^;^t+ [M^m^rn-i.m")dt ' "'^^'W'^S both sides of this equation by M+2.7n. we have Af+r.m. Const. = M. T m.Mzi^ 4- ^. (^''^Z-/^^') , „.. jxVy'-y'-'^VO | ^ ^^.[xdy—ydx-\-x>dy'—iifdj!^ . „ Udy—ydx-'r3^'dy'~u"dsf') mm. ________ I „,,„ _v — £ — J — J J (/< ^ dt A. ^„.'(''dy'-}i'<tx' + x-di/'-y"dx') , {xdy-ydx) , „ (x'dy'-t/d^ dt +»" ^ f-™- rfT H-m"^ i^^^i^-5'-!!^ „, (^cLf-ydx) ,Jxdy'-t/dx') ,„ {x"dy"-y"dx^) J , {jfdx'—xdif) , „{ydx"—xdy") ^ , {y'dx—x'dy) -f- mm'. — i. + mm".^ ; :i— ' + mm'.-^ ; ^ lit dt ^ dt ^^■rr^'.'^'^-^A^!i^^^.,^y:dx-^:d^ ^^ dt ^ dt ~ dt „„,>\idy—ydx-ifx'dy—y'dJ\, , {ydJ—xdy') , Av di —i' dy) . . mm -^ — ^ ——■ — ^ ■ + mm'. -■? = — i-i -\-mm'.^ ; £.'+4c.= dt ^ dt ~ dt . {{x—T).(dy'—dy)—( u'—y).(dx'—dx)') «'«--^ -^-^ y^'y yn —"'')\ ... making the factors of n/m", m'm", *f. PART I—BOOK II. 41 at ^.mm. < ^ — -—^ <-^ > ; (5) [ydz — zdy') c"-=.MX.m.- dt ^ (6) c' and c" being two new arbitrary quantities. If we multiply the differential equation in x, by ^ , „ H.m.dx 2mdx — '2m.- the differential equation in y, by „ , - l.m.dv -~mdy-^m.—^, the differential equation in z, by li.m.dz 2mdz — 2m. and if, in like mannfer, we multiply the differential equation in x', by PART I. BOOK II. G also to coalese, and obliterating the quantities whicl> destroy each other, we have (M+2.m). Const. = c=the second member of tlie equation in the text, it is evident that the same proof is applicable to any number of bodies. 4a CELESTIAL MECHANICS, M+Hm ' the differential equation in y', by 2m'.dy'. — 2ot. ^ ■ the differential equation in 2', by 2'm!.dz'.~'2m\- and so of the other bodies ; if we then add together these different equations, observing that we will obtain A_£>v idx,d'x-\-dy.d^i/ ^-dz.d'z) Q'E.mdx d^x 2I..m.dy ^ d^y £E.m.dz ^ d'z . ^,, rwrfr * The differential equation in x, being mul(iplied by this quantity becomes = + M. — \-2mdx.l.-~- —2A—-\dx — — .m.- -rrr^— ■ dt* mx 2m . »«x , 2 , /^ dx \ .„ — —X.mdi,~—— S.mdx.'S.-—4- rr; .S.mdx.l -r- 1, if corresponduig operations be perfonned on the differential equations in x', x", &c. we will obtain by adding them toge- ther, dxd^x . ,,„ mxdx , „ , mx ^ (dxl . 2z.m. — — -- +M.22. — +22.7nrf*.2.— - —22.-? y > dx— dt^ r* r* \ OS ) PART I.— BOOK II. 4S which gives by integrating const - Z TB Jdx^+dy^-^dz^) —(•E.mdxy—(J:.mdyY-'i'£.mdzy ' ' dt* {M-\.Y..m')dt'' — 2M.S.— — 2A, r or h=MX.mX ^ •' L + dt"" Z.mm'. \ (dx'—dxy+(di/'—dyy-\-(dz'—dzy ^ * " dt^ d'x 2M , mx 2.S.m , mx 2 dt^ M+2.W , r' M+2.W r* ^ M-\-^.m. M+2.M ^t]''"^^ 2M this equation by reducing, and observing that — rrr . 2.wrf'.r.2. 22.»» , mx „ , mx , , , 2 , dx 2.>«rfx.2. — — = — 22.»«(ir.2 , and also that . 2.mdx.^, =0, M+2.m r^ r' M+2.m ' ' * </x becomes 22.mrfx.-v- + M.22. 2S.-? — J- .rfx— 22. Pidx.^.m.— — ; dt* H \dxi dt^ M+2.m if this equation be added to the differential equations, which result by performing corres- ponding operations on the equations in y, if, y'\ &c. z, «', /', &c., observing also that 2xdx->[-iydy-\-'2zdz=2rdr, we shall obtain the differential equation of the text. '• ^S] ■ '^" + '- [|] • <^ + ^- [Zzl • '^'='^' seepage 28. • If there are but three bodies, we have by multiplying by (Af-f-m-j-wi'+m"); Const. .(M+ )«4-»n'+"'") = h; and if we only consider the coordinates parallel to the axis of x, we will have M (mdx^ + m'dx'^ + m"dx"^) -{- {in -}- m' + m"). (mrfx*+»i'c?x * + m"rfx"*)— (w + m'+?)i")rfx+fl'x'+rfx")]s = Af.2.n;i;^+mVx' + m'^rfx''-|-m"-rfx"^+mm'rfx"+»;m'</x'*+mH!"t/x*+ wm"rfx"'+n2'7H"e?x'»4-M'ni''rfx"«— vi'^dx-^—m'^dx^—m" ' </x" * —2mm'dxd3?—^mm"dxdx"—2m'm"dxdx".=MX.mdx^ + wm' (rfx— rfx')*+(7nm'r.(rfx— £fx")^+m'7?i". [d£—djf'Y, =M2.mdx'-i- Z.mm' (di'—dxY ; si- milar expressions may be obtained for the differentials of the coordinates parallel to the axes of z and y, and if to these be added —{^M.'Zm -|-2a) multiplied by M+S.m, we will have the expression in tlie text. .*^ 44 CELESTIAL MECHANICS, h being a constant arbitraiy quantity. These different integrals were already obtained in the fifth chapter of the first book, relatively to a system of bodies which react on each other in any manner ; but consider- ing their utility in the theory of the system of the world, we thought it necessary to demonstrate them here again. 10. The preceding being the only integrals which have been ob- tained in the actual state of analysis ; we are compelled to recur to the methods of approximation, and to avail ourselves of the facilities which the constitution of the system of the world furnishes us with for this object. One of the greatest arises from the circumstance of the solar system being distributed into partial systems, composed of the planets and their respective satellites ; these systems are so constituted that the distances of the satellites fi-om their primaries, are considerably less than the distance of the primary from the sun ; it follows from this, that the action of the sun, being very nearly the same on the primary and on the satellites, they move very nearly in the same manner, as if they were only subject to the action of the primary. The following re- markable property also follows, from this arrangement of the planets and satellites, namely, that the motion of the centre of gravity of a planet, and of its satellites, is very nearly the same,* as if all these bodies were concentrated in this centre. In order to demonstrate this, let us suppose that the mutual distances of the bodies m, m'. Sec. are very small, compared with the distance of their centre of gravity, from the body M. Let x=X-i-x, ; 1/= Y+y, ; z—Z-Vz. ; x'=X+<; y'-Y^y'r, z-Z^z',; &C.; * See Princip. Math. Lib, Ist- Prop.65» PART I.— BOOK II. 45 X, Y, Z, being the coordinates of the centre of gravity of the system of bodies m, m', m", &c. ; the origin of these coordinates, as also that of the coordinates, x, y, z, x', y', z', &c., being at the centre of M. It is manifest that x„ y,, z^, x,', &c. will be the coordi- nates of W2, 7w', &c. relatively to their common centre of gravity ; we shall suppose these to be very small quantities of the first order, in relation to X, Y, Z. This being premised, we will obtain, as we have seen in the first book, the force which solicits the centre of gravity of the system pa- rallel to any right line, by taking the sura of the forces, which solicit the bodies parallel to this line, multiplied respectively by their masses, and then dividing this sum by the sum of the masses. Moreover, we have seen in the same book, that the mutual action of bodies connected together in any manner, does not derange the motion of the centre of gravity of the system ; and by No. 8, the mutual attraction of those bodies, does not alter this motion, consequently, in the investigation of the forces, which actuate the centre of gravity of the system, it is sufficient to consider the action of the body M, which does not belong to this system. The action of the body M on m, resolved parallel to the axis of x, Mx and in a direction tending from the origin is , therefore the entire force which sollicits the centre of gravity of the system of bodies ffi, 7n', &c. parallel to this line, is* — MX.-— and by substituting in place of x and of r, their values, we have « By what has been stated in No. 20 of the first book, it appears that — - dt' Z.m now in the present case 2.m.P=— Afs.-— . for P =—— — . 46^ CELESTIAL MECHANICS, r* - ((X+^.)* + (F+3/,)«4-(Z+^,)») If we neglect very small quantities of the second order, namely the squares, and the products of the variables x,, ?/,, z^, */, &c. j and if we denote by R, the distance \/X* + F*+ZS of the centre of gravity of the system, from the body M ; we shall obtain X _^ X X, (Xr^+I>£f^) * ~~ W^ R' "~ R' a! x'' we shall have the values of -^j- , -777, &c. by distinguishing the let- ters X, y, z, &c. by one, two accents, &c. ; but by the nature of the centre of gravity, 0=S.ff2X,; 0=:S.7m/,; 0=^7nz/y therefore we will have, neglecting quantities of the second order, T.^„ mx ^•^•— MX S.m R^ ' ((x+x,)*+(y+3/,)^+(z+2/)' ^ ^ '^^ ' ^^ ^'' ^^ ^ '> ^ by neglecting quantities very small of the second order, X.(X'+2Xx,-{-Y'+2Yi/,+ 3 '2 Z» +2Zz,r^ + iX-S^^+ Y^+Z')-^=X{X'+ Y'-+Z^)~''—~X.{2Xx,+2 Y>/,+2Zz,)R' + x,{ X» + Y'+Z^ ) ^= (by substituting R' for X*+Y'+Z^) •;p + "^ — (Xx + Yv+Zz.) ^ ,^ mx I MX.'S.m S.mx, rX2.«x,+ y2.n>.v,+Zv.>.z) ) _ _ MX ^^^ ^^^ ^^^ ,^^ ^^^^ ^j. ^j,, «econd member of this equation ranish. PART I.— BOOK II. 47 consequently, the centre of gravity of the system is sollicited by the action of the body M parallel to the axis of a:, in very nearly the same manner as if all bodies of the system were concentrated in this centre. The same conclusion evidently obtains for the axes of 7/ and of z, so that the forces by which the centre of gravity of the system is actuated parallel to these axes, by the action of M, are pT~> pT" * When we consider the relative motion of the centre of gravity of the system about M, we should transfer in an opposite direction, the force which sollicits this body. This force resulting from the action of ttz, m\ ml', &c. on Mf resolved parallel to x, and acting in a direction tending from their origin, is S. — j ; if quantities of the second order are neglected, this function is by what precedes, equal to XS.wi R' In like manner, the forces by which M is sollicited, in consequence of the action of the system, parallel to the axes oi y and of -2, in a di- rection tending from the origin, are F.E.TW . ZX.m ■, and R^ R» It appears from this, that the action of the system on the body M, is very nearly the same, as if all the bodies were condensed in their com- mon centre of gravity. By tiansferring to this centre, and with a con- trary sign, the three preceding forces ; this point will be sollicited pa- rallel to the axes of .t, of _y, and of z, in its relative motion round M, by the three following forces : -<M+S.»j).-^; -(M+E.m).-il; _(M-}-E./b).-^. 48 CELESTIAL MECHANICS, These forces are the same as if all the bodies 7n, m, m", &c. were united in their common centre of gravity ;* consequently neglecting very small quantities of the second order, this centre moves as if all the bodies were concentrated in this point. * The action of m on M resolved parallel to the axis of x = — , ••■ the sum of the r' actions of all the bodies m, m' , m", &c. on M ; = 2.—, = by what precedes •" *.• if this action be transferred to the centre of gravity, with a contrary sign, this centre in its relative motion about M, will be soUicited pai-allel to the axis of x, by the force — (Af-f 2.ni).— — ; now if all the bodies m, m', ??i",' dc. were concentrated in their common centre of gravity, this centre would be acted on parallel to axis of x, by the force — (M-J- 2.m.)X, ••• this centre moves as if all the bodies were concentrated in it, consequently it de- scribes very nearly an ellipse about M, the quantities which are neglected are of the order of the square and higher powers of x, and it is easy to shew, that the aberration of the force, by which the common centre of gravity is sollicited, from the inverse ratio of the square of the distance, is much less than the aberration of the forces solliciting any of the bodies com- posing the system, from the inverse square of the distance. For if tiiere are but three bodies, and if the distance o(the greatest Mfrora the remaining m and m', be much greater than the distance of m from m', then if 72 be the distance of M from the common centre of gravity of m and m', p and q the distances of this centre from m and m', respectively, and 28- the angle which r=p-^g, makes with R, the distance of M from vi, — R — p. cos. ■et, the distance of M from m' =R4-y. cos. ■a, :• the attraction of M on m, resolved parallel to MR M R = -3 = MR{R-^ +3R-*p. cos. zr+6R-^ p^ cos t«ar+&c. ~ — -}- SMp.COS. w , GMp .COS. '■a) , „ . ,1 ^i_ ..• c n/r I i-=r 1- — Pi ■ — '- + &c. ; m like manner, the action 01 M on m, re- R^ ^ R* ' ' „, „ MR M 3Mi7.cos.sr , eAT^'.cos^ solved paraUel to R^-jr; w, = -rp «1 f" /?; ' ~*'"- (/i-j-y. cos. ot)^ R- it-* li* now we know from what has been already established in the first book, that the accelerating force by which the centre of gravity of in and m', is sollicited in the direction of R, is ob- tained by dividing the sum of the motive forces, by which »i and »i' are sollioited in this direction, by 7ii-\-m', :• this force is = to ( t''!^ + __i^'^_ I . -L. = by substitation (.(iJ— ;j. COS. ar)^ ' (yi-fy. COS.ro)^ J lli+m' PART I.— BOOK II. 49 It follows from what precedes, that if there are several systems, of which the centres of gravity are at considerable distances from each other, compared with the respective distances of the bodies of each PART I. BOOK II. H f Mm SMmp.cos.v, ^GMmp-'. cos, 'ct , ^ Mm' S Mm' q. cos. -a J — ^' '^ (mp^ 4-m'<7' )+ &c., the first term gives the law of elliptic motion ; the se- cond term vanishes by the nature of the centre of gravity, •■• the third and following terms are those which cause an aberration from the law of elliptic motion in the centre of gravity. The actions of m and 7n' on M, resolved parallel to R, are respectively -=; , '■ , which become by reducing, -=— -, — ;rr > and if these {R— p. COS. ■vt)^' {R-\-g. COS. z,)*' ^ ^' R^' R- be transferred to M with a contrary sign, the entire force by which the centre is urged, is p-^ . It appears from this discussion that the centre of gravity of the earth and moon describes very nearly an ellipse about the sun ; now a comparison of this expression, with that which gives the action of M on m, disturbed by the action of m' on M and on m, shews that the curve described by the centre of gravity, approaches much nearer to an ellipse than the curve described by m, for the force on m, acting in the direction of R — p. cos. w _ M+m m'.{R— p. COS. ■a) ,f 1 R-j-q.cos.ir) \ ~ (R—p COS. -sry "^ ~" ~r' '■'"■ l{/J+y. cos. ^=)^ r^ )' cos. 6, ( being the angle at which r is inclined to a radius drawn from M to m, this ex- pression becomes by rejecting very small quantities of the second and Iiigher orders, M4-m-l-m' . w'. COS. S , , , . ., , .,. - + rm — , and the last term is evidently greater than (R — p. COS. ■a)'' [R-\-q. COS. -sr) 6Af. COS. *i<r jMT)'-4-m'(7' „,, .. ,. , . ,. , ^^ • w r — • The force which is perpendicular to R — p. cos. •a is equal to R"- ' in-\-m' "^ ' -^ , f R+q. cos. v! 1 !.,,.'"'• sin. t , ^ m'. } !— — — . S. . sin. fc= by reducing ^- rj ; but l r^ (/? + j. cos. w)* J ' (R-\-q. COS. -ay if the force of M on m, be resolved parallel to r it will be = ; = rr , and the ^ (R—f.COS.-sr)* force of M on m' parallel to r =-=: —> '•' the accelerating force on the centre of '^ (R-^-q- cos. try C Mmp Mm'.q 1 1 f Mmp gravity parallel tor= \^R_pJ^_^y-^Rj^^_,,,^^y\;^:;;^^{-Rr SMtnp' . cos. TO Mm', Mm'q SMm'fl. 'cos. ar 1 1 , / _a R^ ir+ 2_ 1 _p-,= because «p-^'y=0. 3MC0S 33- I, J. U u rr— — ,. ("iy^+w'g'); the part of this force which is perpendicular to it disturos the 50 CELESTIAL MECHANICS system ; these centres will move very nearly in the same manner, as if the bodies of the respective systems were concentrated in them ; for the ac- tion of the first system on each body of the second system, is, by what precedes, very nearly the same, as if all the bodies of the first system were united in their common centre of gravity ; the action of the first system on the centre of gravity of the second, will, therefore, by what has been just established, be the same as in this hypothesis, from which we may conclude generally, that the reciprocal action of different systems, on their respective centres of gravity, is the same as if the bodies of each system proportionality of the areas tlescribed by the centre of gravity to the times, and it is evi- dently less than — - — '■ ' — •, See Princip. Math. Lib. 1. Prop. 66. Cor. 3, 4, &c. '" (E+q. cos, ^Y "^ r > ' The distance of the centre of gravity from M differs from the distance of m from M re- til 1 solved parallel to R, by p. cos. ct, = , • r. cos. w. (by the nature of the centre of gravity"). In like manner the abberration m longitude =p. sm. ss- = — — — ;. r. sm. ■a, ••• it varies as the sine of the angle of elongation of M from m ; if i be the tangent of the latitude of the earth, the distance of the earth from the plane passing through M and the centre of gravity m' of »n and m', = sp = rs, r> t\ow «=tan. (f. sin. (v — $), ^ being the inclination of the orbit of the moon to the above mentioned plane, and v — 6 being = to the distance of the m' moon from her node. The distance from this plane, as seen from the M = — - — r- . '^ m-\-m ^. See Book 7, and Newton Princip. Math. Prop. 65, 66, 67, 68. What has been R stated at the commencement of this note, shews the truth of Newton's 65 and 67 Prop. Lib. 1. And it would be easy to demonstrate, as Newton states in Prop. 64, that when the force varies as the distance, the centre of gravity describes an accurate ellipse about M, for the force soUiciting m parallel to the axis of x, = — Mx, ••• the force which solicits the centre of gravity parallel to this axis, — — — — MX '■ ■', now this last terra vanishes, if we add to this force, the force 2.mx = X2.m-fS.m.j;; by which M is sollicited in a contrary direction, the entire force on the centre of gravity parallel to this axis = — (M-{-'S,.m.)X, V the centre of gravity describes an accurate ellipse, and m describes an ellipse about the common centre of gravity of Man d m' ; the periodic time in this elb'pse depends on the number of bodies composing the system, and it varies inversly as the square root of the sum of the masses. PART I.—BOOK II. 51 were concentrated in them, and that consequently those centres move, as they would do, in the case of this concentration. It is manifest, that this conclusion equally obtains, whether the bodies of each system are free, or connected together in any manner whatever, because their mu- tual action does not affect the motion of their common centre of gravity. Therefore, the system of a planet and its satellites acts very nearly in the same manner on the other bodies of the solar system, as if the planet and its satellites were united in their common centre of gravity ; and this centre is attracted by the several bodies of the solar system, as in this hypothesis. Each of the heavenly bodies, being composed of an infinite number of molecules, endowed with an attractive power, and their dimensions being very small compared with its distance from the other bodies of the system of the world; its centre of gravity is attracted very nearly in the same manner, as if the entire mass was concentrated in it, and it acts itself on the several bodies of the system, as on this hypothesis j therefore in the investigation of the motion of the centre of gravity of the heavenly bodies, we may consider these bodies as so many massive points, placed in their centres of gravity. But the sphericity of the planets, and of their satellites, render this hypothesis, already very near to the truth, still more exact. In fact, these several bodies may be conceived to be made up of strata very nearly spherical, and of a density which varies according to any given law ; and we novr proceed to show that the action of a spherical stratum on a body, which is exterior to it, is the same as if its mass was united in its centre. For this purpose, we will establish soiue general propositions, relative to the attractions of spheroids, which will be very useful in the sequel. 11. Let X, y, z, represent the three coordinates of the attracted point, which we will denote by m ; let dM represent a molecule of the spheroid, and a/, y', s^, the coordinates of this molecule, j denot- ing the density, which is a function oi'af, y, z', independent oia;,y,z-y we will have dM zz ^.djfdg.dz'. H 2 52 CELESTIAL MECHANICS, . The action of dM on m, resolved parallel to the axis of x, and tendiqg towards the origin, will be ^,dx'.dy'.dz',(x — x) {{^—'^y-viy-y'y^iz—z'rf^ ' and it will consequently be equal to ^ J ^.dx'.dy'.dz' ^ {. dx } therefore if V denote the integral p ^.daf.dy'.dz' ^^ J y/{x-xy-^iy—j/Y^^{z^^Y ' e.dx' .di/ .dz' * The action of dM on m, is expressed by , rrrr} — TvTTT ^ » ■•" ^^^ force p.di!.di/ .dJ . , paraUel to the axis of ^:^^_^,^._^^^_^,^,_^^^_^^. : : (^-x ): ^.dif .di/ .dif •^{x-xr+{y-y'Y+^-^r . consequently it is =^(^_y).+(_yly).+(,_^),)i . »!« p,dx' .dy' .d:! expression ,-, ,^, , ,~ yx. , , iTxTT' differenced with respect to x, and divided by p.dx'.dy.'dz'Jx—x') .. dx. becomes— ^j^__^,;,^^^_^,,)r)^ : V th^ express.onor r , e.dx'.dy'.dz' -i 3 a. ,, ,, , /,, ■ , ,., y , expresses the actidn of a molecule of the sphe- l ^/ix—x)^+(t/—i/y+(.z—zy 5 dx roid, on a point without the surface of the spheroid, consequently, if we take the sum of the corresponding expressions for all the molecules of the spheroid, «. e. if we take }d f ^' ^' — > = — ■( -r f I this quantity expresses the dx action of the spheroid, on the point m, resolved parallel to the axis of «; the characteristic, d refers solely to the coordinates x, y, t, it does not denote an operation the reverse of that indicated by the characteristic yt PART I.— BOOK II. 53 extended to the entire mass of the spheroid ; — < —— ?- will repre- sent the entire action of the spheroid on the point vt, resolved parallel to the axis of cT, and directed towards their origin. V is the sum of the molecules of the spheroid, divided by their respective distances from the point attracted ; in order to obtain the attraction of the spheroid on this point, we should consider F" as a function of three rectangular coor- dinates, of which one may be parallel to this line, and then take the differential of the function, with respect to this coordinate ; the coeffi- cient of this differential, affected with a contrary sign, will express the attraction of the spheroid parallel to the given line, and directed to- wards the origin of the coordinate to which it is parallel. Denoting the function ((4: — a/)*+(?/— ^')*+(^ — 2;')*)"% by S, we will have As the integration only respects the variables a/, y, z', it is manifest that we will have but we have ^-{dx'S'^idT/* S^ i^J'' • ^ — (^— x') d»e _ _i J 3(t— /)t _— (^— r')'— (v— v')*— f2— «')'4-3f*— y^' 54 CELESTIAL MECHANICS, consequently we will have also This remarkable equation will be extremely useful in the theory of the figure of the heavenly bodies ; we may make it to assume other forms, which will in different circumstances be more convenient ; for instance, let a radius be drawn from the origin of the coordinates to the point at- tracted, which radius we will represent by r, let 9 be equal to the angle, which this radius makes with the axis of x, and w the angle which the plane passing through r and this axis, makes with the plane of the co- ordinates X and 7/ ; we will have a; = r. cos. 9 ; i/ =r, sin. 6. cos. w ; z = r. sin. 6. sin. w ; consequently we shall obtain by means of these expressions, we can obtain the partial differences of IB like manner, -— , -— - , are respectively equal to d'Z d-S . d'<i rfx* t/y* dz- -3(«:-'y-3(j/-^0'-3(g-/)' +3{ji-:^y+S{y-y'y + S(z-zy _ ^ * PART I.—BOOK II. 3S r, 6, and w, with respect to the variables x, i/, z ', from which we can deduce the values of ^-^-^^ /—r-iiA-rxi* ^" partial differences of V, with respect to the variables ?•, 0, and zj. As we shall have occasion frequently (o consider these transformations of partial differences ; it will be useful here to trace the principle of them. V being considered first as a function of the variables x, 7/, 2, and then, of the variables r, 6, and w, we have i dx * In order to obtain the partial differences, \ — > A-t\ i\-^\ t it is only ^ idxS IdxyidsS necessary to make x the sole variable in the preceding expressions for r, cos. 9, and tan. -sr, consequently, if we difference these expressions, we will have {rfr "> . C 6?9 > sin. 9 d-sr ^|=cos.9;|^|.= —'>-^=0; by substituting '"'■ { ^ C ' { ^ } > we obtain the value o*" | ^ } » which has been given IN the text. 56 CELESTIAL MECHANICS, which gives ' m=-'-m- sin. 0 (dV .4^1 By this means we can obtain the partial difference 1 —r- c , in partial differences of the function V, taken with respect to the variables r, 6, and V. By differencing this value of^-y— C a second time, we shall obtain the difference \ -^ ^ in terms of the partial differences of V, taken relatively to the variables r, 9, and n-. We can obtain, by a si- milar process, the values of -5 —^ >, and'S . ^ r* By the preceding operations, we can transform the equation (A) into the following : ^ Cd'F? cos.0 (</F) C dT > , U\rV} ,n^, sin. *( _ _ ■/y'+z' . j <^'* 1 _ 2xVJ/^"+2^_ 2.sin. 6. COS. ^ . J ^ 7 . 7* ' '■ l'rfx« j ~ r* ~ 7^ ' Ida )'* tcfx^i ~ *^ ' '■' 1 "^ ) "~ a!) « * dx* "^ dr'lx^'' di'- 'cfx'"^ d6' dx^^ d*V dV sin. »« , d^V sin. «« <fF 2sin.«. cos. « • COS. *P+ — . -^— -I • ; dr- ' ^ dr' r ^ (/O" * r» ^ </« r« C dr 7 y . frfV) 1 v' xMz* COS. *«+sin. »». sin. '» -{|}. Bin. *. = -.^,v{|-J =^^^p^=^, bysubsUtuting for - sin. » its PART L— BOOK II. 57 ralue ; and by substituting r. cos. S for x, and r. sin. ^. cos. ■d- fqr ^, we obtain, r '^ COS. i. COS. -KT d^i 1 X x^* 2xy' cos. « cos. 9. cot. '« {</'«!__ x_ xy^ 2xy _ cos. « 1^ 3 — V'^M^-»'*~~ (^^+~')'-'"' V^y-f;'Jr*~ sin.«.r' 2cos.«.sin.<.cos. 'ot f f/sr T , tan.w r r,, , «* 1 sm. «. r 1 +tan. ■'•cj un sin »in. IP cd^-a ■) 2j/z _ 2. sin. v. cos. a- f ^ 1 C dV \ { dr •* m.i.r' Xdy^l ~{y^+z^Y sin. "«. r' ' ''1 dy \~ \1? J *l'^ J . (dV\ (d6-\ (dV ^ Cd-^f fdV} . ^ , CdVi + U]'\Ty\+{-dz}'{Ty\={77\-''^-'-'^^^--+{-di}' COS. S. cos. ■a CdV "h sin. ar .. f'^'^ \ r \rf^ j ' rTsmTT' '' \ dy^ j Crf^O C«f"^'7 C'^'''? . C^^^l id^'u,^ d'V . , , dV ldpi+id^yiW'l'^\d^\'\^'s=^''^- '■'''• "^-i;^' cos. '^+ sin. '<■ sin, 'u W'F? cos, ^g. cos, 'ct S'^^? ^ cos. <— cos. ». cos. V 2 sin. <. cos, i. cos. ^1^ ^ rf' F sin- *«t dV 2 sin. sr. cos. «r c?r « ___ P J '^ d-a-'- *">». sin. ^T "5^* FTsmTTS ' 1^~T'" >in. *«. cos. *Kr ( «?0 7 »- is? (••■"•' . , . d'r 1 z» x*+2/' COS.**. +sin sm. 4. sui. «r; -; — = = ^— = dz" r r^ r^ sin. *. cos )s. *. sin.CT _ S^^\ cos. *. sin. w ^■^ ^ rf'O "^*£__ COS.* COS. *. sin. 'iff V^'-f«'.r»'^(i(»+z»)-Tr» V^»^z».r* r^ sin. « sin. «. r» 2 sin. S. COS. «. sin. 'ar fcfw'} ^ cos. w rf'w 2zv r* ' \dz J "" ^*+s« ~ r.sin. «' dz^ ^*+T^ ~" 2sin.ti7. cos.g fc?F-) ^dV y S'^f\^SdV\ WHj.f'^^l V'^'l sin.»<.r« '\1^\-\TrV\Jz\^ \di\'\Tz\'^ XdZl'lIz^ PART. I.— BOOK II. S» CELESTIAL MECHANICS, if COS. 6 be put equal to i*, this last equation will become f(^ri . CdV\ COS. fl. sin. ST . idVt cos.w id'-V\ = | — I .m...sin..+ |_} . 4.|_}.__;|_} C<^'^? ■ 2, • « , Cc^^? Ccos.««.+sin. »«. COS. »o7 , td'V\ = J^}- -=«•««'• *- + [^^.^ S- l + ilFi' COS. '*. sin. '■!? , C^^ 1 COS. « — COS. «. sin. '■KT 2sin.fi . [. \-r- > i-T-— J— COS. fi. sin. 'w • r* ^ «/« J r^sm. « r' iT. COS. w f'^'^l COS. V 5^1 2 sin, ^j. "*" l"rf^J *PTsmT« Irfari* rSsin.*« ' if the corresponding terms are made to coalesce in the values of i -— j + ^ -7-7 \ + I -^ ? , we will obtain the following expression C d''V ■) dV r sin. ^6 \ —5- ^ . (cos. ""^^-sin. *«. COS. *«r^sin. '^6. sin. *«)+ -r-.) — '■ \r cos, "fi.+sin. '<. sin, 'ct . cos. M+sia. ^L cos, 'ct \ ^d^ Vl sin. a<+cos. 'fi.cos. *g-+cos. ^L sin, ^ig) J <^^ 1 J 2 sin. «. cos. fi cos. 6 ") ^dV\ f2sin. «. c r*sin. ^ cos. (. COS. *iiT 2 sin. i. cos. fi.cos. 'w cos. 0 cos. fi. sin. ^-a ' r». sill, fl ^ " "*" r^. sin. i r'.sin. < 2 sin.fi. cos.fi. sin, 'ct^ j^ ( rf^T i sin. 'w cos. *sr 1 _i_ J '^^' 1 "~ r» 5 I. <^a-* J r^ sin. "fi J-', sin. »fi j i ^ 3 (2 sin, ig. cos. OT — 2 sin, tg. cos, zr^ _ frf^Fi nf^^l ^j.)*^^^! ^ r'.sin. »fi )~ t'^^l''" {"dV I'T l~^J *T^ , CdF-» cos.fi ((i^F 1 1 , , . , . . -r \~jT \—7—- — -+ \ -rr- f • -r— — rr=0, V hence multiplying by r«, we obtain I rffi ) r>.sin. fi l dw^ i r».sin. *fi tr j a j ' 1 rfr» J ^ I dr i^ 1 c/fi» ) ^ i dfi 5 sin.« l^'w' i sin.ifi "•""'"'' PART I.— BOOK II. 59 12. Let us now suppose, that the spheroid ,is a spherical stratum, the origin of the coordinates being at the centre ; it is obvious that V will only depend on r, and that it will not contain /ut or w ; the equa- tion (C) will therefore be reduced to from which we obtain by integrating, r I 2 \ ~1F' S^^'-Xd^S'^^'X'dP]''^'' •'^"'S considered as constant, •.• r. | -^ J may be substituted in place of r». s ? + 2r. ■{ — f . Ifwe make COS. . = ^. then —= (_).(J^), and _ = (_). -^ +f -^ V f ^ J) and as rf« is constant, and rf^= — rfO. sin. «, d*fi=: — dS*, cos. *; d'V .d-^V. dV. ,iV. ,dV^ rf^ dV j/<^^\ cos.<_^ dV ^ '/l—f^Kft. _ /rf*FN ,rfF. cos.»_ rf*y . sin.«~ <'/**'/lZ:^ * W«'/'^('5r}' sin.« ^ <//*»* ^ ''*' -j-y /« = ''({1 — t** ^-7-^ ; hence it appears how the equation (B) may be re- dft duced to the equation (C). t If the attracting body be spherical, the quantity V will be always the same, when r h 60 CELESTIAL MECHANICS, A and B being two constant arbitrary quantities. Consequently we have from what precedes, it is manifest, that - — ^—rc expresses the action ( dry of the spherical stratum on the point m, resolved in the direction of the radius r, and directed towards the centre of the stratum ; but it is evi- dent, that the entire action of the stratum must be in the direction of the radius; therefore — ]~r~i expresses the total action of the spherical stratum on the point m.* First, let us suppose this point to be placed within the stratum. If it was at the centre itself, the action of the stratum would vanish : therefore when r=o, we have — < — r- = 0, i. e. — = 0, from \ dr } r* the same, and it only varies when r is increased or diminished. For suppose the attracted point to move on the surface of a spliere, concentrical with the attracting body, it is evident that the value of V remains the same when the attracting body is spherical, but when this body i» any other figure, V will vary from one position to another of the point moving on the spheric surface. K-apr ) = 0. V -^ = ^, and r F= Ar+B, it appears from this equation, that if r^O, B^O. • From what has been stated in page 42, relative to the action of a spheroid, it ap- pears that — (-7-/ expresses the action of the stratum parallel to r, but it is evident that the entire action of the stratum is equivalent to this expression, for if equal elements be assumed at each side, equally distant from the direction of r, their action perpendicular to r will be destroyed, and the remaining action will be in the direction of r, and this being the case for every two corresponding elements, it is true for the entire spherical stratum. PART I.— BOOK II. 61 which it follows that B=0, and consequently whatever may be the value of r, — -| —r- >= 0 ; from this it appears, that a point situated within a spherical stratum does not experience any action, or, which is the same thing, it is equally attracted in every direction. If the point m exists without the spherical stratum ; it is mani- fest that if we suppose it at an infinite distance from its centre, the action of the stratum on this point, will be the same, as if the entire mass was collected in this centre ; therefore if M represent the mass of this stratum ; — s— :— r or — will become in this case, equal to — r-» from which we obtain B = Af, therefore we have universally,* r' * When the point is at the centre — j- = 0, when r = 0, as has been already re- marked, see preceding page ; this is also evident from other considerations, and as B must he the same, wherever the point is assumed within the surface, B in all such cases s= 0 ; V V=A, the value of A may be easily determined. When the point is infinitely distant, the action is the same as if all the molecules were united in the centre of gravity of the sphere, see page 47, and in this • • , M idV) B M D ,T i^ ^ . -^^ case the action IS equal to , v — \ — S- or — = , ••• B=M; V=A-\ . hence when the attracted point is infinitely distant, A=0, •.• it is always =0 ; and V= r r ' If the attracted point be without the sphere, the attraction towards the convex part is equal to the attraction to the concave part of the surface : and when the point is on the surface, the attraction to the spherical stratum is only half of what it is, when the point is at a distance from the surface. This is immediately evident from the expression u*.du.d'a.d6.sm,i ^^ ,., , , „ 1 , . j j j^ • « -jr [r — u. co».«. «.(/)), which, when ip-if)OC-— becomes u*.du.dv!.di. sm, t. T'^^ti, COS 6 ' ' , and it is easy to shew that this expression is the same for two elements situ- ated on the convex and concave sides of the spherical stratum, and which lie on two lines drawn from the attracted point, and making an indefinitely small angle with each other, for u sin. tz=. a perpendicular let fall on r from the attracting element, r — «. cos. 6 = 62 CELESTIAL MECHANICS, with respect to exterior points, :dV} M CrfF7 _ M 'id? i ~ r* ' that is to say, they are attracted by the spherical stratum, in the same manner, as if the entire mass was united in its centre. A sphere being a spherical stratum, of which the radius of the in- terior surface vanishes ; it is obvious, that its attraction on a point si- tuated on its surface, or beyond it, is the same as if its mass was united in its centre.* This conclusion is equally true, for globes composed of concen- trical strata, of which the density varies from the centre to the surface according to any given law ; for this is true for each of its strata ; thus, as the sun, the planets, and the satellites may be considered, very nearly, as globes of this nature ; they attract exterior bodies almost, as if their masses were concentrated in their centres of gravity, wliich is conform- able to the result of observation, as we have seen in No. 5. Indeed, the figure of the heavenly bodies deviates a little from the spherical form ; however, the difference is very small, and the error which results part of r intercepted between attracted point and this perpendicular, and it is manifest from similar triangles that the perpendicular let fall on r, and also the intercepts between these perpendiculars and attracted point are respectively as the distances of the attracting elements from the attracted point, and udS is also in the same ratio in both cases, see Princip. Math. Book I. Prop. 72, •.• for the two elements at above mentioned, u.di.u. uu.. 6.(r — m.cos. ^) . , r ■, , . .1 .• i.- i_ ■ IS the same for both, consequently the attractions which vary as these expressions will be equal, and this being true for every two corresponding elements existing on the same right hnes, itis true for the entire stratum. Hence if the attracted point is indefinitely near to the spherical surface, its attraction to the molecule contiguous to it, is equal to its attraction to the rest of the spherical stratum ; if the attracted point ap- proaches still nearer, so as to become identified with this molecule, it will then be a part of the stratum, and its attraction will now be only half what it was previous to its contact with the stratum, * For w being the radius of the homogeneous sphere M= -rr— • "'» V — < -j- f — PART I.— BOOK II. 63 from the preceding supposition, is of the same order as this difference, relative to points contiguous to this surface ; and with respect to those points which are at a considerable distance,* the error is of the same order as the product of this difference, by the square of the ratio of the radii of the attracting bodies to their distances from the points attracted, because we have seen, in No. 10, that the sole consideration of the great distance of the attracted points, renders the error of the preced- ing supposition, of the same order as the square of this ratio ; the heavenly bodies, therefore attract one another very nearly as if their masses were concentrated in their centres of gravity, not only because they are at considerable distances from each other, relatively to their respective dimensions ; but also because their figures differ little from the spherical form. The property which spheres possess in the law of nature, of ac- tracting, as if their masses were united in their centres, is very remark- able, and it is interesting to know whether it obtains in other laws of attraction. For this purpose, it may be observed, that if the law of gravity is such, that a homogeneous sphere attracts a point placed with- out it, as if the entire mass was united in its centre ; the same result will have place for a spherical stratum of a uniform thickness ; for if we take away from a sphere, a spherical stratum of a uniform thickness, we will obtain a new sphere of a smaller radius, which will possess the property equally with the first sphere, of attracting as if the entire mass — j-= when r=:a, — — . a; for a point which is situated within the sphere, it is evident the action of the strata between the point and exterior surface vanishes, consequently this case is reduced to the former. • This ratio may be deduced from what has been established in No. 46, page 10; sec also Systeme du Monde, page 255, and Book 3, No. 9. If the force varied as the distance, a homogeneous body of any figure will attract a particle of matter placed any where, with the same force and in the same direction, as if all the matter of the body was collected in the centre of gravity. See notes to page 50. This will appear immediately if the force of each element be resolved into other forces parallel to three rectangular co- ordinates. 6* CELESTIAL MECHANICS, was united in its centre ; but it is evident, that if this property belongs to these two spheres, it must also belong to the spherical stratum which constitutes their difference. Consequently the problem reduces itself to determine the laws of attraction, according to which a sphe- rical stratum, of an uniform and indefinitely small thickness, at- tracts an exterior point, as if the entire mass was collected in its centre. Let r represent the distance of the attracted point from the centre of the spherical stratum ; u the radius of this stratum, and du itg thickness. Let 9 be the angle, which the radius u, makes with the right line r, -u the angle made by the plane which passes through the two lines /■ and m, with a fixed plane, passing through the right liner; m"c?m.c?ot.c?9. sin. 0,* will represent the element of the sphe- rical stratum. If then f denote the distance of this element, from the point attracted, we will have f^ = r*—1ru. cos. 9-f«*. Let us represent the law of the attraction, at the distance /"by <f{f), the action of the element of the stratum, resolved parallel to r, and directed towards the centre of the stratum, will be , , , ,. . , (r — u. COS. 0) . ^x udu.dis.dM. sin. ^.- -7. . '^\J)\ but we have r — u. COS. 9 ~{^P / in consequence of which, the preceding expression assumes this form • The three sides of the element, are du in the direction of the radius, udi the ele- ment of the curve in the plane passing through the radius u and r, and u sin.<. dm the element perpendicular to this plane ; see Book 3, No. 1. PART I.— BOOK II. 65 u\du.dzr.dl sin. 9. J ^ | . ?>.(/ ) ;• therefore if we denote J^djl (?(/), by (p,(J) ; we shall obtain the entire action of the spherical stratum on the point attracted, by means of the integral t^.du.fdvs.d^. sin. ^'p,{f), diiferenced with respect to r, and divided by dr. This integral relatively to w, should be taken from t3-=0, to n- equal to the circumference, and after this integration, it becomes ^Tt.u^.du.fd^. sin. 6. ip/y) ; TT expressing the ratio of the semi-circumference to the radius. The value ofy differenced with respect to 6, will give TU and consequently, 2,r.«Vw./d/9. sin. G. ^X/) = 27r.i^ • ffdf. <?,{/). PART I. — BOOK II. K * The attraction in the direction o^ J" :z: ti^du.dvr.di. sin. t. <p(J'), and as r u. COS. 6 =s the distance of the attracted point from a perpendicular demkted from tfae at- traoting element on the direction of r, it is evident that u^du.d-a.di, sin. «.?{y). • '— — — is equal to the action of the attracting element in the direction of r, _ _ ,r dy "l _ r — u. COS. 9 ^ c?/._ tf«. sin. * 66 CELESTIAL MECHANICS, The integral relative to 6, must be taken from 9 = 0, to 9 = 7r, and at these two limits, we have J'^z r — u, andy z: r + u; consequently the integral relative to j^ must be taken horn /"= r — u, to J'= r+Uy therefore let X/W- <?,(./) = ^ (f) > ^^'^ shall have* -ffdf. <p(J) = . 3 t^r+w)— ij.(r— w)^. r r L J The coefficient of dr, in the differential of the second member of this equation, taken with respect to r, will give the attraction of the spherical stratum, on the point attracted, and it is easy to infer from thence, that in the case of nature in which <p(X) = T^"'^ *^^^ attrac- * The action of the entire stratum, in the direction of rzz.v.'^du.fd-a.di, sin. i. < -^ I , Q [f) =: u^dic./d-a.di. sin. 6. " ' =: u^du./dia. di. sin. 6. (p,{J) differenced with res- pect to r, and divided by dr, dr and di being independent variables. The attracting force for each molecule = ti,''.du. dzr.di. sin. *• ] ^ f • <?( /)) *•' in order to obtain the entire force a triple integration is requisite, with respect tajl to 6, and to sr. In order to integrate with respect to di. sin. 6. ?i, (y"), this expression is reduced to a function of y only, and as _/" is here considered as a function of 6 only, r comes from under the sign of integration ; by substituting for di sin. 6, we get 2wu^du.J'di. sin. i. ^If) = -^ .duffdf, (p,{f), and a.%df\s only concerned as far as/ is a function of t, and as the limits between which the integral of the first member of this equation ought to be taken, are 6—0, 6=7r, to which limits the corresponding values of/ are r — u, r-j-"> i. e. the least and greatest values of/, it is evident that by makiDgJ'/df.(p,[/) = Mj')> the integral of the second member will assume the form in the text. t <?(/)=j5. ■■'/df. <?{/) = Hf)^-jr' and//i/:oX/)=^K/) = -/=at the limits, — r—u, +r—u; ••• i|/(?-+2i)_4(r— ;() = — 2k, consequently, the differential PART I.— BOOK II. 67 tion is equal to — ^^ . that is to say, it is the same, as if the en- tire mass of the spherical stratum was united in its centre j which fur- nishes a new demonstration of the property, which we have already established, on the attraction of spheres. Let us now determine <?(y), from the condition that the attraction of the stratum is the same as if its mass was united in its centre. This mass is equal to 4:Tr.ti"du, and if it was collected in its centre, its action on the attracted point, will be 47r.w*c?M.(?(r) ; therefore we shall have o J S d.]—.{mr-^u)~^(r—u)-^\ ^ ,. , . ,-p,. 27r.ttf/M.< L r ^^ 3 > = 4!Tr.vrdu. ?(/•)} (D) and by integrating with respect to r, we shall have 4'(r+u) — »]^(r — u) =:Qru.Xdr. (?{r) + rU, U being a function of u, and of constant quantities, added to the in- tegral* ^u.fdr,(p{r). If we represent ^(r-\-u) — ■^(r — u), by R, we shall obtain by differentiating the preceding equation. id'R\ , , . , „ d.(p(r) (. dr-j dr K2 coefficient of the second member of this equation, with respect to r= — '— — . {—2?/) = ; iiru*dtp= the mass of the spherical stratum, for 5ra'= the area of a circle whose r' radius=«, •••■ixu' = the surface of the spherical stratum, and ^jtu. 'du= the mass of the stratum, of which the tliickness =; du. * Multiplying both sides by dr, and dividing by 2ic.udu we obtain by integrating 68 CELESTIAL MECHANICS, but by the nature of the function R, we have t~d^s - Cd^V'^ consequently, or o ^^ ^ ^ . r.d.<p(j)-> (d"Ul 3<r-) d.<p(r) _ 1 ^d'U iu'idu' V r dr 2m Thus, the first member of this equation being independent of u, and the second member being independent of r, each of these members t For ffdj. <Pif)^^f), -.-fdj. <f,{f)=d. ^(f), and df^<pji/)->rdrf. ?(/) = d'Mf), •■• (dr+duy. (<?,(r-\-ic)+(r+u). ^(r+u))= d'4{r+u), {dr ~duY (?),(r_jO + {r—u).q>{r-u))=d\i,{r-u); ^_d\-^{r-^u)—d\-^{r—u) _ d^R __ d\4.(r+ii)—d'4{r~u) _ d'R dr' ~ dr' ~ dti" ~ du' ' In order to obtain the attraction to a sphere, we should integrate the expression ' W'^-i-^) — ■4'('' — u) from m=0 to u = L, L being the radius of the spliere, and then the differential of this function taken with respect to r, and divided by dr, will give the attraction of the sphere. — See Book 12, No. 2. PART L— BOOK II. 69 must be equal to a constant arbitrary quantity, which we will denote by 3 A ; therefore, we have + — -7- — — j^ , r dr from which we obtain by integrating, B (p{r)z=Ar+-^i B being a new arbitrary quantity. Consequently, all the laws of at- traction, in which a sphere acts on an exterior point, placed at the dis- tance r from its centre, as if the entire mass was collected in this centre, are comprised in the general formula Ar-\ — -. In fact, it is evident, that this value satisfies the equation (Z)),t whatever may be the values of A and B. If we suppose A zzO, we shall have the law of nature, and it is evi- dent that in the infinite number of laws which render the attraction very small at great distances, that of nature is the only one, in which * Since u does not occur in the first member, nor r in the second member of tliis equa- tion, the equality of these members can only arise from their being respectively equal to a constant quantity, independent of both u and r. Multiplying both sides by r*dr, we shall have 2r.ir.?)(r)-|-r'.rf.(p(r)=3^r'.rf>-. •.• r'.<pr= Ar^-\-B. t In this hypothesis fdf(p(f) = A.fd/.f+ B/ -^ = ^ - y- = H/)- and 70 CELESTIAL MECHANICS, spheres are endowed with the power of attracting, as if their masses were united in their centres. And if a body be situated within a spherical stratum of a uniform thick- ness throughout, it is in this law only that the body will be equally at- tracted in every direction. From the foregoing analysis, it appears that the attraction of a spherical stratum, of which the thickness is ex- pressed by du, on a point placed in its interior, is equal to ^ dr ^ In order that this function should vanish, we should have ^{ii-{-r)—^(u — r) = r. U, U* being a function of u, independent of r, and it is easy to perceive (y4+4.r3M+6r'M''44rM^-t-M'*) — B{r-l-u), and n}^(r— u)= — . (r*—^r^u-\-6r''u''— ■ira'+u*) — B[r — u), :• ^{r-\-ii)—-^{r — u)— A.[r^u-\-ru^)~'2Bu; and d.i-. {i'ir-YtC^—Mr—u) \ = d. i^.A{r^u + ru^)—'2Bu)\ dr dr 3Ar3u+Au'r—Ar^u—Au3r4-2Bu „ ^ . 2Bu = •■ I ■ . . ■ ■ =2^rM -f- — ; r' r'- and if we substitute for (p{r) its value Ar •{ , in the second member of the equa- tion (D), it comes out equal to 2Aru-\ j-. • U being the constant arbitrary quantity which is introduced by the integration of PART L— BOOK II. 71 tliat this is the case in the law of natvire, in which 'p(J') zz-r:^. But in order to demonstrate that it only obtains in this law, we shall re- present by ^'(f), the difference of 4'(/)> divided by df; we shall likewise denote by V(X)> ^^^ difference of ^'(f) divided by df, and so on ; we shall thus obtain by two successive differentiations of the preceding equation, with respect to r, ^"(u+r)—^\.'Xu-~r) zz 0* As this equation obtains, whatever may be the values of ti and r, it follows that ^"{J") mu^t be equal to a constant quantity, whatever may be the value ofj"; and that therefore 4'"'(y)= 0} but, we have by what precedes, from which we deduce d, — (%|.(«-|-r)— ^)fl— r)), differenced with respect to r, if ■ ■ ■ w only equBl to U, its differential with respect to r must vanish, for then the quantity to which this JO differential is equal vanishes : ue- 4wM\rfK^r=0. ^^ <P{f) ^^^-jiif^f* 9(f)=^^j(-/)~ — — , mdfdff(pXf} -—/Bdf- — B{f), :• ^u Jg. r)~^(u-r) = B.(—r—u)- CI ") 9Sr 3.{—u+r)= —2Br; -.' d.>—. ■^[uJf-r—^u—r) S = — rf. = 0; r is B.(—u-\-r)=. —^Br: •.' d.i—. ^(u-i-r—d^lu—r) > = — rf. = 0: r is less IT than u when the point is assunied within the sphere, •>• the limits of/ must be taken ji+f, u — r. * d-M-+-)^-M—r) ^ u=,^'{u+r)-nu-r) : and r{^^r)-V(«-r) = \ 72 CELESTIAL MECHANICS, and therefore 0 = 2. K/) +/ <pXf) J which gives by integrating, ?>(/) z: — ,• and consequently the law of nature. 13. Let us resume the equation (C) of No. 11. If this equa- tion could be generally integrated in every case, we would ob- tain an expression for V, involving two arbitrary functions, which could be determined by seeking the attraction of the spheroid on a point situated in a position which facilitates this investigation, and then comparing this attraction with its general expression. But the inte- gration of the equation (C) can only be eflFected in some particular cases, such as when the attracting spheroid becomes a sphere, in which case the equation is reduced to one of ordinary differences j it is also possible, in the case in which [the spheroid is a cylinder, of which the base is a curve returning into itself, and of which the length is infinite : we shall see in the third book, that this particular case involves the theory of the rings of Saturn. Let us fix the origin of the distances r, on the axis itself of the cy- linder, which we shall suppose to be indefinitely extended on each side of the origin. Denoting the distance of the point attracted, from the axis by r', we shall have r' = r.v/l— ^*. dp - rfr ~''• lJ/'(^t+r) is always equal to \J-"{m — r), now this could nqt always be the case unless each of them was constant. • M/) =ffdf. 9JJ); :■ ^>{J) =/. ^,(/), and V{/) = <P.(f) + /• K/), and +"'(/)= ?(/) + ?(/) +/<5'(/) = 0, multiplying by fdf^e obtain 2f<!>(f)df + /'n/)-4f^0, •.■fK<p(f)z=B, and <?(/)= ji' PART L— BOOK II. 73 It is obvious that V depends solely on r' and -r, because it is the same for all points, i-jlatively to which, these two variables are the same ;* consequently it only involves f/., inasmuch as /■' is a function of this va- riable ; which gives thus, the equation (C) becomes, PART I. BOOK ir. L • / . — a perpendicular let fall fi-om the attracted point, on the axis of the cylinder, t = the angle which ;- makes \vith the axis, •/ T^=r. sin. t=r. V' 1 — ^^« ; if the base of the cylin- der was circular, F would be always the same, when / was the same, i. e- it would be a function of r' only, but as this curve may be an ellipse, or any other curve which returns into itself, F must depend also on the angle which the plane of x, y makes with the plane passing through r, and the axis of x, i. e, on w. , TfiMfi dr.dfcft dr.dfifi r.dft.' '^ VI— jK»' ^/l— ^» VI— jk"' VI— ;«» (1-^^)4' '■■ ■^~ (i-^»)i '''d;^~id?^\'[d^\'^\d/r Sd*r'\(d-Vl ,, ,^ ^ i'^^l , d^V . (d'.rVl ^ , ^ . . [TP^in^r (l-''')-2^- {^ } +_:5l + ''- i -rfF- 1 = ^^ (subst.tut,ng 1-^- 7* CELESTIAL MECHANICS, from which we obtain by integrating, Vzz <p(r.'. COS. OT+r'. \/ — 1. sin. t3-)+i]/(/. cos. -st — r'. \/ — l. sin. ■ar);* ■•• '^'■•'•^•=7p^- dV^^.d-V; hence r.|^ | =-^. ,Vl_^«+r'. ( 1 — ft' ). ^ -—J ^ . By substituting these values, the equation (C) becomes 1— i" (fp 2r.(l— fc') ^ ; . — ^ — ■ — j^ :=. 0, and if both sides of this equation be multiplied by 1 — fi*, we wiH obtain the expression given in the text, by substituting }' for r.v/l — fc'. d^V d^V * This integral may be deduced a priori in the following manner : let ^ := r, ,^ — t, = o, then we will have r-\-r'-.t-^r'.q = 0; the general expression Rk* + dr Slc+T=.0, Lacroix, torn. 2. No. 752, 753, &c. becomes i'+Z'^rO, ••• k = ±r'.\/—l, antldu= — r. (di-' +/c.rfsr), dv=—r-r- (d>' + k'd-a) become by making ——, — — = dr dr dr dr" respectively — p, and substituting -j- s/ — \.r', — v' — i./, fori and A'; du= — + V'— 1. rfcr, dv= —j v' — 1. dvT, consequently K=log. Z-}-*^ — l.tn, i= log. r'—^ — 1. PART L— BOOK II. 75 ?(?•') and 4/(r') being arbitrary functions of r', which may be deter- I, 2 . , du 1 dt w, are particular integrals of the preceding differential equations ; let —y = — 7 = n; -r-, 1 , du ^j, = -7~" ' TT" ='^— 1. = m; —- = — V— 1. = m ; g = np' + nq', (see Collection of r/ q' Examples of differential and integral calculus, page 466,) = —■ +— y-S *" = — ""z "1" aZ—p'—o'+fl'-ffl' =0, -.• 4/=0, i.e. 4^ = 0, and F = <p'(«)+^(u) =^ ip'(log. /+ •^n t^)4- iJ.'(log. )■'— •liT. w))= respectively, ((?' log. r'+log. e""*'^— ^•)+J''(log.r'— —^VZT log^ ^=?''(log. (r'.lcos. «7+i/— 1. sin, w))+^'((log. r'.(cos. ro— V— 1. sin. w)) — ip{r'. cos. jT-fr' V — 1. sin. to) + 4'{>^' cos. ar — /. %/ — 1. sin. w), by substituting cos, w ±v — 1. sin. «r for e~ ^ , and assuming the arbitrary function <p = the function ip'. log. This integral evidently satisfies the preceding equation, for / ^"\ _ d.(p{r'. cos. w+r'.V' — 1. sin. ■a) d.(r'. cos, isi+r'y — 1. sin, ■a) ^ '''■ / rf.(/;cos.a-+rVZ:f sin.sr) ^^^ ^^ '^•('•'•cos CT— r'.V^— 1. sin, to)^ ^ (/. cos, to— / V^^ sin, p) tf.(r'.cos. TO— ?-'.\/-IT. sin. ar) dr' rj_E^_ q".i?.(/.cos. to4->^.\/— 1. sin.TO) rf.(/. cos.TO+r'.^— 1. sin, to)'- ^'^''" ^ d.{y. cos. ar+;^ ^Hr. sin. ■^)» * dr"* rf. ^(r. cos-TO+Z-y/ — 1. sin, to) </'.(/. cos. to+/V — 1. sin, to) (/.(r'.cos. TO-f-rV-ir. sin.TO) * ^'''* ■ t^^4-(/.C0S.TO — /.\/— l.sin ;s-) d.{r'. cos, ar— /.y/I^. sin, to)' (/.()-'. cos. -sr—r'.V—l. sin. to)* * d)'' ,'- , d. Mr' cos. TO— /.v^— 1. sin, to) c?*.(/. cos. zr—r'.'/—l. sin, to) rf.(r'. COS. TO— /,\/ —1 . sin. to) 76 CELESTIAL MECHANICS, mined, by investigating the attraction of the cylinder, when ■a- is equal to cipher, and when it becomes equal to a right angle. but rf.(r'. COS. tiT±r'V — 1. sin. 157) , / — - . — i -y-. =COS. •sr±V — 1. 3in. w, .*. dr d'-.^r'. COS. ■srit.r'. ' — 1. sin. ■iit)_ dr" i Sin. ■a) (dV\ of.ifi(r'.cos. OT+r'.^/— l.sin. w) ,, , , , :rj ) = —^ ,— '-. {/. cos. m+r'.'/—l. "■^ -^ d(/.lcos. OT+r'.V — l.sin.is-) rf.-4-(r'.cos. w — /.v — 1. sin. Iff) , , , , . + —^ p= '. (/. cos. «7— r' .V— 1. sin, ■a) diy. cos. w — /.v — 1 . sin. w) , ,d^V. d'.i?(r'.cos.iff+/.\/IIi.sin. 57) ,, , , ^, — . . , v r'^.(—-.\ — i — = '. /*.(cos. *ar+2V— l.sin.w.cos.»— Sin.ijr) \dr'^ } d(r'.cos. w+jV_l.sin. ^)^ , rf».4'('"'- COS. sr—r'.v' — l.sin. jsr) ,, , ^, — - . . , , -\ — —^^ ^^.r'*.(cos. 'ot— 2v — 1. sin. «r. cos. is— sm. 'w) </.(/. cos. zr — r' . V — 1 . sin. <aY (AX\ — d-<p{r' .co%.vs-^r' .*/ —\ . sin. -0) d.jr'. cos. ■a+r'V — I. sin, to) ''"'' rf.(r'. COS. sr+r'.v'iri. sin. w) ' ^ rf.4-(j-'. COS. CT — r'.^/ — 1 . sin. ■sr) rf.(r'. cos. w — /. \/ — 1. sin, ■p) rf.(j-'. COS. ■a—r'.^—\. siii. ot) "'" J'r c?'.(p(/. COS. ar+r'.y/^T. sin, to) d.(r' . cos. ^+r'.\/—i. sin.ar)' (/to= "£?.(/. cos. ot+Z-'*^-^. sin. to)» * ''■='" fl?.(?i(r cos.TO+r'.\/— 1. sin, to) rf'.(>-. cos. TO+r'.\/ — 1. sin, o-) rf (/. cos. TO-f /. ^/ — 1. sin. to) d^T^ , d'^.^{r'. cos, a- — r'.y/ — 1. sin, to) rf. (y-'. cos, to — r^.v* — 1. sin, g")^ (/.(/. cos. TO— r'. \/Iir. sin. ar)* * ^^^ rf. ■v|'(r'.cos. ar — r'.ij — 1. sin. to) d'.{r'. cos. sr — r'.^/ — 1. sin. to) rf.(/. cos. TO— r'.v/— 1. sin. to) ' '''»* PART L— BOOK II. 77 If the base of the cylinder is a circle, F will be evidently a function ofr', independent of 13- ; the preceding equation of partial differences will consequently become, which gives, by integrating, t dr'i - r' ' <^.(r'.cos.«r±/V— 1. sin. jsr) , . , , , — - — i r = — /.SUl.W±/.v' — 1.C0S.W. aw dK(r'. COS. u±r'.'y^r. sin. zr) , , ,—-.. , (tl'S -T-j = — r.cos. wrpr. v' — l.sin. w); •.• \ ^„^J rf'.«)(»^. COS. ■CT + r'.\/ — 1. sin. w) ,. , . , ^ , — - . , , = ■ ■ ; . r'».(sin. ^■a—'2V — 1 .sin. w. cos. «r — cos. 'w) d.{y, COS. «r+r'.\/ — 1. sin. ot)^ , ^.^'(r'. COS. 3-4/. \/ — l-sin. w) , ,, , ./— T • \> 4» —^ . (-^ j'.(C0S. zr-f- V — 1. sm. -a)) d:{/,cos, •sr + r'.v' — l.sin. w) ^dK4^(/. COS. ^-V^. >^.sin ^)^ ^^ ^^.^_ '^,)+2v/=T. sin. «. cos. «_cos- U). d.{r'. cos. -a — >/ — 1, /. sin. *ot) rf.il/(r'. cos. TO — /.V — l.sin. w) , , , ./ — r • v« , — ii ^= '— . (_r. (cos. -n— V— l.sin. to)). rf.(r'. cos. ar — r'.^. — 1. sin. to) substituted ; consequently this integral satisfies the given differential equation. When TO Tanishes F= ^(r')-\-^{r'), and when to=:90°, V=<p(t'-»/—\)-\-^—r'.'/—\), and as the attraction in the direction of r' = -j -rr- > , ?(/), and ij'(r') may be determined. 78 CELESTIAL MECHANICS, H being a constant quantity. In order to determine it, we will sup- pose / very great with respect to the radius of the base of the cylinder, which consideration permits us to regard the cylinder as an infinite right line. Let A represent this base, and z the distance of any point of the axis of the cylinder, from the point where r' meet this axis, the action of the cylinder supposed to be concentrated in its axis, and re- solved parallel to r', will be equal to f- Ar. dz the integral being taken from ;:= — oc, to ^ = oc ; which reduces this integral to — — ; this is the value of — \—r-,>j when r' is very consi- derable. By comparing it with the preceding expression, we obtain H = 2 J, and it is evident that whatever may be the value of r, the 2A action of the cylinder on an exterior point, is — —.* * If the base of the cylinder be circular, V will be always the same, when / is given, ••• V will be a function of r', independent of w ; dividing by /, and multiplying both sides by d/, we obtain r = >/r'*-\-z'; .'. the attraction in a direction perpendicular to the base, : to the at- traction towards the assumed point = — — ; — - ) ;;/ : v^/^+z', hence as Adz is the dif- A'/dz ferential of the area of the base; — — ? is the differential of the entire force and its [r +z )'^ Az integral = . /-,^ =, (see Lacroix, No. 192), when z = OC this integral becomes A A — , and when z = — OC, it becomes — —r ; and as we want the attraction of the pomt r' r to the cylinder between these two values of z, the difference of the expressions in these PART I.— BOOK 11.^ 79 If the attracted point lies within a circular cylindrical stratum, of an uniform thickness, and of an infinite length ; we have also — ) — -i 2A two cases, = — — > must give the attraction required. Wlien r is very considerable with respect to the radius of tlie cylinder, it is the same thing as if the mass of the cylinder was concentrated in its axis. When the point is situ- ated within the cylinder, F is of a different form from what it is, when the point is situ- ated without the cylinder ; and as it is of the same form wherever the point is assumed witliin the cylinder, whatever it is in one case, it will be the same in all. The length of the cylinder must be infinite, otherwise the point, even when situated in the axis, would not be equally attracted in the direction of the axis. When the base is circular, — ) — - i = —7- •.• — | -— i . clr — H. — —, • • V (^<tr y r \ dr \ r =H. log. r'-j- C. The cylinder being of an infinite length, the attraction perpendicular to the axis is the only attraction which it is necessary to estimate. Therefore the force varying inversely as the square of the distance, there are two cases in which a point is equally attracted in every direction ; the first is when the point is situated in the interior of a spherical stratum, (it will be proved in the third book, that this conclu- sion maybe extended to the case of elliptic strata, the interior and exterior surfaces being similar, and similarly situated ;) the second is that in which the point is situated in the in- terior of a hollow cylinder, whose base is circular and length infinite. If the cylinder was concentrated into a right line of a finite length, the attraction in a direction perpendicular to this line = — — ^ — —I. of which the inteijral is / ^ ^ (j^ + z'^Y ^ V(r'+z^)r'. And if a is ^ the length of this line, the entire attraction in a direction perpendicular to it " ... . . 1 — y , ,; hence if a be infinite, the attraction is as — ; the attraction in the direc- z zdz tion of a, is as ;-— — —3 ; •.• the differential of the force = —r-n — rr^* the integral of which -1 1 1 1 IS ,- , . ; + C, when 2^0, C= — - , •.•the entire attraction = -— — ' Vr'-\-z^ " ' r' ' ■ / K^r^+z" — — 7==^r=- = when z ■=. a; ; ; •.• the attraction in the direction of a is to the attraction in the direction of r' ::</ r'^-\-a'^ — r' : a\ hence it is easy to de- termine the direction in which the point would commence to move ; it may be easily 80 CELESTIAL MECHANICS, :r — ;- ; and as the attraction vanishes, when the attracted point r is on the axis itself of the stratum, we have H zz 0, and consequently shewn that a point placed in the vertex of a triangle is attracted towards the segments made by the perpendicular with a force reciprocally proportional to the secants of the angles which the base makes with the sides. For if r' be the altitude, and a, a, the segments of the base, it is evident from the expression -; — that the attractions to the segments (I a' are as / = to — r- > but these expressions will be evidently pro- V'a^-l-r'"- Va'^+r'' portional to the reciprocals of the secants of the angles at the base of the triangle. If the attracted point exist in a perpendicular to the plane of a circle which passes through the centre, x being the distance of the attracted point from the circumference of a circle, con- centrical with thegiven circle, the distance of the centre from this point being=;-', then ■xr.fjc'^ — r'*)=the area of this circle, and ^.■xxdx is the differential of the area, and as the attraction in t' the direction of r' b as — j- ; the differential of the attraction of the point towards the circle iTT.r'.dx „,.,,. , . IW ^ , , , , =r , or which the mtegral is 1- C, and when x = r the attraction va- x'^ X ■ishes, ••• C=r 2a-, and the corrected integral = 1t.(\ I, hence the attraction of a point situated in the vertex of a cone to all circular sections of the cone is the same, and for similar cones the attraction varies as the side of the cone. If the attracted point exist in the produced axis of a finite cylinder witli a circular base, of which the radius =«, r' being as before the distance of the attracted point from any point in the axis, \/n''-f»"'* will be the distance of the circumference of the cylinder from this point, the attraction to- r wards this circumference is as I — , , and the differential of this attraction is as a, ''' ,-^ ,^ of which the integral = r* — v'a'l-)-'^, r, and r^, being the greatest and least values of/, the attraction to the entire cylinder = — r, + /,, — v'a»+r^„ -j- ^a^-f-r/; r, — r//= the length of the cylinder. If the length be infinite r, =V^a*-f-r/', •.• the attraction is as r, — •</ a^-\-r^, and if a be infinite the attraction is as r, — r,, , the length of the cylinder. PART I.— BOOK II. 81 a point situated in the interior of the stratum is equally attracted in every direction. 14. We may apply to the motion of a body, the equations A, B, and C, of No. 11, and then elicit from them, an equation of condition, wlrich will be found very useful, in verifying as well the computations of the theory, as also the theory itself of universal gravitation. The differ-- ential equations (l), (2), (3) of No. 9, which determine the relative motion of m about My may be made to assume the following form : dt- ~ I dx)i ' df ~ IdyS' di^ ~ IdzS' *^'^ „, . ,^ M-{-m ^ m'.(xx' + 7/u' + zz') a , • • Qbemg equal to E. ^^ ^j^ ^ + — ; and it is easy to perceive that we have :d-'Q) (d^Q) {d'Q 0 = dx % provided that the variables x, y', z', x\ &c., whfch Q contains, are independent of x, y and z. PART 1. BOOK II. M \dx\ (x"+^^-|-z )4 "• r-^ ■ m'\dx\'' \dyS (j" + i/« + z*f -^•"TT + -;;r 4 5^ r i "S 5 = - F+T+T)! -^•— + — • i 5; 1= ''"* mx mV mx rf*x Mr mx 1 dx« (x^+3^H2^)t (x'4-_y^-|-2*)l 7/t'lrfxM ~ /_ — m^ %m.{x—if . 8i2 CELESTIAL MECHANICS, The variables a; y, z, may be transformed into others, which are more convenient for astronomical purposes, r being the radius drawn from the centre of M to that of m, let v represent the angle which the projection of this radius on the plane of x, and of y, makes with the axis of x ; and 6, the inclination of r on the same plane ; we sh^U have, X z=. r. cos. 6. cos. V; y =. r. cos. 9. sin. v ; z — r. sin. 9. By referring the equation (£) to these new variables, we shall have by No. 11, \d'Ql .„ UQ) , id-Q} UPQ) sin. 6. UQ) O = r«. dr' \ " 'IdrS'^ IdvA "^ MGM cos. ^.' Id^V^ ^ cos. ^9 Multiplying the first of the equations (i) by cos. 6. cos. v ; the se- cond, by COS. 9. sin. v; the third, by sin. 9; and then, in order to abridge, making . d*r r.dv* „. nc?9* iVi'=: :lj._ — ^j:ri_ . cos. ^9 — dt' dt' ' '"" " dt* ' — "' Sm'(y' — yY \4.Ar (x'-:r)^ + (i>'-y)^M^-zy)\ + ((x'-.r) •+(y-^) '+ {z-zf ) "** f — »/ %m'(z'—zY ,\4.&c rf'Q rf'Q (j'Q _— 3(M+ffl).r^-f3(M+OT).r'' — 3m^(J'-I)'4-(/— y)H(z'— z)-)^ ((x'— .r)- + (»/-t/)^ + (z'-^)')' -|-3ffl'.,; , (, 7/3^.Vl +';--) s :=0. In the expression for -.W + ;^ + VTi PART I.— BOOK II. 83 we shall obtain, by adding them together. ' dr In like manner, if we multiply the first of the equations (0. by — y. cos, 9. sin. v ; the second, by r. cos. 6. cos. v, we shall obtain by their addition N' being supposed equal to </.j /'«.—— . cos. *9 (. di Finally, if we multiply the first of the equations (Oi by — r. sin. J. COS. V ; the second by — r. sin. 9. sin. v ; and if then we add them to the third, multiplied by cos. 9, we shall obtain, by making P' equal to , rf«9 , dv^ . ^ , ^r.dr.d^ r^.—, hr*. . sm. 9. cos. 9 + dt' ' dt* dt' ' Id^y are only considered the first terms in each, but as the other terms are precisely of the sauie form, it is evident, that the sum of the three differential coefficients, for each of the other terms respectively constitute a result equal to cipher. * dx ■=! dr. cos. 6. cos. v — dS. r. sin. 6. cos. v — dv. r. cos. i. sin. v ; '.• d'^x = dH. cos. *. cos. u — dr.di. sin. i. cos. v — dr.dv. cos. 6. sin. v — d'-S. r. sin. C. cos. i' — di.dr. sin. 6. cos. v — di^ . r, cos. 6. cos. v -f di.dv. r. sin. 6. sin. u — d^v.r. cos. 6. sin. v — dv.dr. cos. i. sin. v -|- dvM. r. sin. i. sin. v — dv'^ .r. cos. 6. cos. v, •.' d'^x. cos. 6. cos. d = d*r. cos. '^. cos. *t> — 2rfr. «(<. sin. 6, cos. S. COS. -v — 2dr.dv. cos. ^S. sin. v. cos. i' — d^i, r. sin. <. cos. 6. cos. 'v — (/<'■. r. COS. *«. COS. "ii-|-2c?i;.rf«. r. sin. (. cos.^. sin. r. cos. v — d^vr. cos. *^. sin. u. cos. v — rfu'. r, cos. »«. COS. "^u ; di/=: dr. cos. «. sin. v — rd6. sin. S. sin. v -\-rdv. cos. «. cos. v; :• d'!/=d^r. COS. i. sin. II — rfnrfS. sin. 6. sin. v + dr.dv. cos. «. cos. u — dr.df. sin. <. sin. v — rdS'^. cos. J. sin. v—rdLdv. sin. «. cos. v — rd'K sin. d. sin. v\-dr.dv. cos. *. cos. v — rdv*. cos. ^. sin. v— rdv.de. sin. ^. COS. V -|- rd'' v. cos. (. cos. v; v d^i/. cos. 6. sia.v = d'^r. cos. '(.Bin. 'v. 84 CELESTIAL MECHANICS, The values of ?; v, and 6, involve six arbitrary quantities, which are introduced by the integration of the preceding differential equa- — Qdr.de. sin. 6. cos. 6. sin. ^v -{■2dr.dv. cos. ^6- sin. v, cos. v — rd$^. cos. '6. sin. 'v — rd^t. sin. <. COS. (1. sin. ^T) — rrfn'.cos. ^6.s'm.^v-i-rd'v.cos. 'i. sin. r. cos. v — Src'^.rfr.sin. *. cos. *. sin. t). cos, u; dz—dr. sm.6-\-rd6. cos. fl ; •.• d'z—d^r. sin. 6. -\-2drdS. cos. 6 -\-rd^e. cos. i — rdi^. sin. S; ••• d^z. sin. 6=d'r. sin. '^6-[-2dr.dL sin. ^. cos. 6-\-rd'(. sin. L cos. ^ — rdt'^. , rf^x (i*u . , d'z . d^r ids'" sin. I 6, consequently, -j-^cos. 6. cos. v-\- -~ cos. 6. sin. v-|- —3- sin. S=-t-i —rj- ctt ctt ('i (*(• at rdv- ^ dx dy . dz . d'x ; — . cos. '6, but -r— = cos. (. cos. V ; — p- :=cos. «. sin. ii ; -7- = sin. S. •.• -—— dt^ dr dr dv dt^ , '^'i/ . • a- '^^^ • . COS. *. COS. 11 + , . cos. L sin. v-j- — -r-. sin. ^= di^ dt^ In like manner, if d'x and its value be respectively multiplied by the differential of x, on the hypothesis that v is the only variable quantity, we shall obtain ; — r.d'^x. cos. *. sin. 1)= rd'r. cos. 'i*. sin. v. cos. vf2dr.de.r. sin. d. cos. S. sin. d. cos. v-^2dr. dv.r. cos. =e. sin. ^v+d^6. r" . sin. u. cos. r. sin. ^. cos. i-\-di'^.r'^. cof, '^ sin. t). cos. v-\-d~v. r». cos. *#. sin. 'u+rfi)' r-. cos. ^<. sin. v. cos. u — Idv.di. r^. sin. «. cos. 6. sin. 'r ; and multiplying d^y and its value by the differential of y, taken on the same hypothesis, we obtain r,d''y. cos. *. cos. v—r-d'r. cos. *^. sin. v. cos. i) — 2c?r. dS. r. sin. ^. cos. i. sin. ». cos. D_j_2rfr. dv, r. cos. '^. cos. ^u — r'^di'^. cos. »*. sin. v, cos. d — r*<f *S.'sin. i. cos. ^. sin, v. cos. v — r'. <fo*. cos.' S. sin. 1). cos. t'4»'^rf^u. cos.'^. cos.^u — 2r'^ .di.dv. sin. *. cos. S. cos.*u; .• r.d^.'- cos.f_ sin. 'j+rd'y. cos. «. cos. t)zr2rcfr<^t). cos. '^6+y^d^v. cos. *« — 2r*rfu. dS sin. (i. cos. *. „ ^ dx . dy C (^^x 1 = (/.(r*.(fv. COS. '^); -5-= — »• cos «. sm. u; -j- := r. cos. <!. cos, v, -.• — i -^ > . r.cos.e.sin..+ |^|. r. cos. .. cos. . = J-j. j— j + |^j. |^^ — i — \:=zN'. Multiplying d^j and its value, by the differential of x, taken on the supposition that 6 is the variable quantity ; — rd'^x. sin. 9. cos. v= — rd'r. sin. *. cos. I. COS. 'v+2rdr.d6. sin. '«. cos. 'ii4-2r(/r.rfu. shi. 6. cos. «. sin. v. cos.v + r*d6^. sin. «. cos. t. COS. *w ^r'di.dv. sin. '«. sin. ij. cos. v + rVu' sin. «. cos. 6. cos. ^v + r'^.d^i. sin. '«. COS. 'D+r'd'v. sin. «. cos. 6. sin. r. cos. v. ; performing a similar operation on d^y and its Talue, we obtain —d^y.r. sin. 6. sin. v^—rd^r. sin. «. cos. «. sin. ^ 5i-j-2) rfr.(^<'. sin.^i. sin. *« — 2r<ir.c/v. sin. 6. cos. «. sin. u. cos. v ■\- r^di^. sin. ^. cos. 9. sin. 'v -j- 2r^.d6. dv. sin. '<• PART I.— BOOK II. 85 tions.* Let us consider any three of these which we will denote by a, b, c ; the equations M' = \ — { will furnish us with the three fol- ( dri lowing equations : idr ]'\da§'^ Xdr.dvyXdaS \dr.d^]'\da} ~ \ da J' \dr-yldb}^ \dr.dvS'XdbS "*' Xdr.dl^yidb^ " t db > ' \ dr'S'ldc J "*■ Idr.dvS'ldcy ^ Idr.dri' ldc\ -~ I dc V We can obtain by means of those equations, the value of ^ , a y and if we make idv\ cd^^ cdv^ rdn "'={Tb\'{d-A-Uj'{dby' '' = uru}-{da}'U\' sin. v.cosiv -J- r'dv^. sin. 6. cos. f. sin. 't'-+- r^.d^S. sin. 'S. sin. 'v — r'd'v. sin. *. cos. i. sin. V. COS. v; and in liiie manner d'^z.r cos- 6:^rd''r. sin. 1 cos. tf-j-2rrfr. dS. cos. '< — r'^dt'. + , , , (/-.T.r . d^ii. r . . , d^z.r T^d'^L cos. ^, •.• T— sin. ^. cos. V ^^-^- sin. «. sni. t) H = — dt^ dt^ dt^ ' 2rdr.de r'^.dv^ . d') , dx . \dy cos. i— \- sin. 6. COS. 6-\-t ".— — , but — — = r. sm. 6. cos. v ; — r- = — r. sin. «. sin. v. dz , ( rf*«"l J^V ] • ■ . f '^'^ 7 — = cos. « ; and — < -r-, f . r. sni. «. cos. v — < -r-^ > . r.sin. 6. sin. n-J- < -rrr f ^-cos. «. '2r.dr.d6 d-6 . (dv"] r: j-T r-'' --I— sin. «. cos. fl+r«. -{—-!- = P. di" ~ dt^ ^ \dt^ i • The vakes of r, v and 6 are determined by the integration of equations of the second order, ••• two arbitrary quantities are invoWed in the determination of each variable. 86 CELESTIAL MECHANICS, ^' - UayidbS LJbS' Id^V' - ldc]'W'^dc^~^d'J'^dc^'^dby' '^ ldhl'\dc^'\da^~^Jbl'lTa^'\Tc^' [drl {dv\ ff/6 7 Ulrl (idvl \d^\ ^dc^'^la^'^Tb^-l'ckl'idb^'^d'J' we shall have * From the value of J — I = M' ; it is evident that M' is a function of r, v and «'; »nd as these coordinates are functions of a, b, c, and conversely, it follows that I da I 1 dr \- IdaS'^ \ dv \' ida]'^ \ dS \'\da\~ fhy substituting for M' its value -I -j- \ \ Xdr^ r \da i"^ Xdr.dvS' 1 «'« j I dr.dS ]' \daS ' by similar operations we obtain the values of j —jj- }• • \ —j- \, &c. Multiplying J — — I and its value, by m and its value, J.— jr-J and its value, by n and its value, ' dM' •% ^ —j— i and its value, by p and its value, we obtain /rf'Ql idr\,(dv\ (di\ (dv\ f"'M ^ . ^ ''"'^ PART I.— BOOK II. «7 Ib like manner if we make (.drl Sdn Sdrl S '^^ I '''= Ida^' idc^-lTc^'ld-ay' Sdrl Sdn idr} ^dn P'= lTbyiTa\-\TJ'Uby' iTc\-{dc\-U\)=''"iiir\- id-a\-id-a\'id-c\)'='n-drl Sd^\ s^lfS^'l /^\_/£!\ f 'l!.\)4. S JIB} X dr^ ]'ldcS ^ Idal'Xdb] \db]'\daP^ Idr.dvy {dv\(idv\ Cd0-) Cdvf (dCW, i dQ\ ^dil ( <:dv1 \dcS^\da]'ldb\- IdbS'Xda])^ ld7MriToy\lday id-bS~idbS'id'aS)^^'t'd^y Adding these three expressions together, and observing that the coefficients off J , I — — ) are respectively equal to cipher, and that the coefficient of ( ) = S, we will obtain the expression given in the text. We can by a similar process obtain the values •f f——\, ( -T-j )i now if we substitute these values in the equation (F), and also M' and ^'' for ■( J- f ; i -J- ( > 3"d multiply by € and cos *e, we will arrive at the equa- tion (G). 88 CELESTIAL MECHANICS, the equation N' -rzf —\ will give Finally, if we make Idby IdcS Idcy ldb)i ' ^dr-) cdv-i cdr) Cdv') Ldcy Cda^ CdaS' tdcS' P" \drn \dv) (rtr^ S"'^? ~ id^iS' (M) ~ ldh\ ' Idal' The equation P^— -s -j^ r will give Consequently, the equation (F) will become, 0=wi.r* cos.«8. 5 —7—( +n.r'^ cos. *e. < —rr- >-\-p.r^. cos. '9.3^ f ^ da ' t db J I dc y + rw". cos. 9\ \ — ^ + n". cos. ^^-{-tA + /'• cos. '9. | -^ | . + S(fZrM'. cos. ='^— P'. sin. ^. cos. ^). In the theory of the moon, we neglect the perturbations, that its action produces in the relative motion of the sun about the earth, which implies that its mass is indefinitely small. Then tlie variables a/, y', z', which are relative to the sun, are independent of j:, y, z, and the PART I.— BOOK II. 89 equation (G) obtains in this theory ; it is therefore necessary that the values found for r, v and 9, should satisfy this equation, which fur- nishes us with a means of verifying these values. If the inequalities which are observed in the motion of the moon, are the result of a mu- tual attraction between these three bodies, namely, the sun, the earth, and the moon, the observed values of r, v and 6, deduced from obser- vation, should satisfy the equation (G), which furnishes us with a means of verifying the theory of universal gravitation ; for the mean longitudes of the moon, of its perigee, and of its ascending node, occur in these values, and a, b, c, may be assumed equal to these longitudes. In like manner, if in the theory of the planets, we neglect the square of the disturbing forces, which we are almost always permitted to do; then, in the theory of the planet, of which the coordinates are ,r, 7/, z, we can suppose that the coordinates x', yf, z', x', &c. of the other planets, are relative to their elliptic motion, and consequently, independent oix^y^z; therefore the equation (G) obtains in this theory.* 15. The differential equations of the preceding No. drr rdv' — T-s-. cos. *9 — r. — -= }——i dt- • dt^ Idr^ de d.Cr^.——. cos. ^9) ,„ , ^ dt -S^Q) J.; (H) dt Xdv^ di' ^ df ^ dt' I d& S PART I> BOOK II. N * We arrived at the equation (G) on the supposition that x,j/, z were independent of a/ w', »', &c. In the case of elliptic motion x, y, z, are independent of x', y', z', and conversely, and as when the square of the perturbating force is neglected, the motion is q.p. elliptic, it follows that x, y, z, are in this case independent of x', y' , z' . See page 49, of the text. 90 CELESTIAL MECHANICS, are only a combination of the differential equations (?) of the same No. ; but they are more convenient, and better adapted to astronomical com- putations. We can assign other forms to them, which may be useful in different circumstances. Instead of the variables r and 9, let us consider u and s, u being equal to -, that is to unity divided by the projection of the ra- dius vector, on the plane of x and of y ; and s being equal to the tan- gent of 6, or to the tangent of latitude of m above the same plane, by multiplying the second of the equations (H) by rdv. cos. *9, and theu integrating, we shall obtain I ti.dtS -^ \dv \ u^ h being a constant arbitrary quantity ; consequently we have dv dt — '•V*-.^/{f}.^" If the first of the equations (H) multiplied by — cos. 6, be added to the third multiplied by — '- — , we shall obtain u 1 dx^ , idQ) , ^ ^dQ) ^ u df Idu^ IdsS dt from which we deduce There are two distinct objects, one to verify the values of r, v, 6, and the other to verify the theory of universal gravitation. dv /do:^ ,, , ^/. dv /dQ.\ PART I.— BOOK II. 91 (u'.dt) u.dt C ( du) u (ds)J If we consider dv as constant, we shall obtain by substituting for dt its value, which has been already given ~ dir . jdv } ti'dv du u" \ds\ * Cay ) vr N 2 rfV </t)' , rf«« rf«« rf,,i =rfr. (X)S. S — rdi. sin. 9; -.'rf'. — =:d^r, cos. 0 — 2dr.dd. sin. ^. — rf'^. r. sin. 6 rdi^. cos. «■ •.• by concinnating and substituting — rf*. — , for its value, and noting that r.f— ) (dv'^\ fdv'\ . 1 dv'' -—). COS. e — J-.l -J— ) . COS. 6. sm. ^6, we obtain —d^.^ 4- r.—r- dt^ ^ ^df- ' u '^ dt^ • cos de fdQ\ f^(i\ sin. i du (-), .„. * = _ ...c». ,. ...- (f). COS. ,=(f ). ... CO.. .. (f ) = (f ). /ofQ\ /du\ sin. « /rfQ\ ,rf.j» sin.S fdQ\ . , /<;Q\ £i:i:i;sin..=^;andL=;^v(f).(l+.0— = ('?)•«. and .ak- r Vl-t-s* r vl4-«2 ^ds ' ^ ' ' r ds ' ing the two cofficients of(- ) to coalesce, we obtain d^ i '"-r- =M'.(sin.««+cos.'C). dt' 92 CELESTIAL MECHANICS, In the same manner, by treating dv as if it was constant, the third of the equations (H), will become (^ — j + us\ — ) . Substituting for dt we obtain Mt- \/'.-w(J«). ^) H- i,..V-w{^^)4 \dv J' u^ fZu H— I -1 I — > , V dividing by dv, and the radical du) ^ u\ ds i quantity we obtain the expression which is given in the text. ds , ^ d^s 2sds^ . \+s^ , d'i d^s 2s '1+s" 1+s^ (l+s'Y u' ' dt'~ u-dt' (l+s^) ds^ , . , s ^ dv'- . ^ . s dv' ^ , 2sds , (sin. 6. COS. 6 =z , ••• r^. —, — . sin. 6. cos. 6= — -— - : 2rdr= ,c-.dt' ' ^ l+s' dt~ u^ • dt' ' u' ^du.(l+s') „ , , 2s-ds^ 2du.ds , d*6 r' A^i 1 r" ^—^ — , V 2rdr.d6 =z „ ; but r^. = — .d.i — c = — ;-. M» ' {l-^s-y u^ '' dt^ dt t-dt^ dt J — '—— f ; •.• by substituting for d'0, d6 and r* their values already given, and d't for ; — its value dt^ .,-.V.^./{g}.A^Jgi.V;..+v]fj-.^ dv PART I.— BOOK II. 9S Therefore in place of the three difFereutial equations (H), we shall have the following : " - "rf^ "^ '' """ Idv /• u'dv " \du J u'Xds 5 By making these equations to assume the following form, we avoid fractions and radicals, (K) +" •-r•-;- -. V, the third equation (H) becomes = s dv* dv.'Uh^ r d's 2s.ds^ Cds_ 2u.du 1 X^TdF^ (l+«*).M^flfi* ■ ■*" i a'* dv.dt J 2.^s' 2duAs_^Uai \'t\^\^3\yi\\ = oy substitut- _— .p^ + _ . 2«rf«.p' + -,. ^. </.+ .-^-r-t- T+T?. -^;;^ — 2rf«.»d^.p'_ $"^1 «a+i'^-Sj-.(14-,»); equal evidently to the third equa- tion (K). + 91 CELESTIAL MECHANICS, ^_d-t , ^du.dt „ fc?Q| dt\^ "*" /r tldvi 'ic.dv~ \duS m * | </s ) 3 ' (L) +iv- Uv \-dv\-''n-du\ -^'^^n-ds\v By making use of other coordinates, we might form new systems of differential equations ; suppose, for example, that the coordinates x and y, of the equations (i) of No. 14, are transformed into others, relative to two moveable axes situated in the plane of these coordinates, and of which the first indicates the mean longitude of the body m, the second lying perpendicular to it. Let x, and ?/, represent the coordinates of m, relatively to these axes, and let nt + i denote the mean longitude of m, or dv^.—r- ; and by substituting -r-, - — , * By differentiating the first of the equations (K), we obtain d't — 2du.dv J , '^Q dividing by dv^ ; we obtain-p^ = — . u\\- J- , in the second and third equa- ^ ' dv^ udv^ dv3 lav i tions, the second should be multiplied by the denominator, and then divided hyh-, the third should be multiplied by the denominator, and afterwards divided by A^M^ PART I.— BOOK II. 95 the angle which the moveable axis of a;,, makes with the axis of a: ; we shall have x=x,. cos. (nt+t) — ^^. sin. (nt+i) ; y=-X^. sin. (jit-\-i)-\r y ,. cos. (n^+ 0 ; from which we collect, on the supposition that dt is constant, d'^x. COS. {nt-^C)-\-d^y. sin. {nt-\-i)—^x—n^x,. dt^~^ndy,.dt; d^y. COS. {nt+t)~-d''x. sin. (nt^ri')-=d^y,—n-yf.dt^-\-^ndxM. By substituting in Q, in place of x and of y, their preceding values, we will obtain This being premised, the differential equations ij) will give the three following ; df ' dt IdxS \dQ\ '-d~y:. d^yj „ _ dx^ ^dQ-) - df Idz S' * dxzzdx,. COS. (nt-{-i)—dy^. sin. (n<-f f)— nx,.dt. sin. (n<+£)— nj/^.(?i. cos. (n<-ft)- dy=dx^. sin. (n<4.e)-j.rfy^. cos. (n<+s)H-nx,.</i. cos. (nt+i)—ni/,.dt. sin. («<+i). d^x=d'x^.cos.(nf+6)— ti*^^. sin.(n<+6)— 2»diB,.rf<. sin. {«<+0— 2«fi?5/,.</f. cos.(«/ + t). —n'^x,. dt^. COS. (n<+e) + w*J/'.di'. sin. (k<-[-s). d\y=d^Xr sin. (wf +6)+c/^?/^, cos. (nf+s)+2«(/j:,.(/<. cos. {nt+,)—2ndi/^.dt. sin. (««+f) — ?^'a:_.c?i^ sin. (?!f-f-s)— n'y,. r/i*. cos. (ni-fe). V d^x. cos. (nf+s) + «?»«/. sin. (ni+i):zd''x,—2ndi/^.dt—n^x,.dt^. 96 CELESTIAL MECHANICS, After having deduced the differential equations of a system of bodies subject to their mutual attraction, and also the only exact integrals, which we have hitherto been able to obtain, being determined ; it remains for us to integrate these equations by successive approximations. In the solar system, the heavenly bodies move very nearly as if they were only subject to the principal force which actuates them, and the dis- turbing forces are inconsiderable ; we are therefore permitted in a first approximation, solely to consider the mutual action of two bodies, namely, that of a planet or of a comet, and of the Sun, in the theory of the planets, and of the comets ; and the mutual action of a planet and its satellite, in the theory of the satellites. We will, therefore, commence with determining rigorously the motion of two bodies which attract each other ; this first approximation will conduct us to a second, in which we will consider the first power of the disturbing forces ; af- terwards we will take into account, the squares and products of these forces; and proceeding in this manner, we will determine the celestial motions with all the precision which the observations admit of. d*i/.cos.(nt+i)—d^a:. sin. {nt-}-t)=d^!/^-\-2ndx^.dt—n\i/,.cit^. = _.„.,„,+„•.■ {^«} = {f}.coM..+.)-{|}..in. (.<+.). x^—x. COS. («i-fO+3/- ^'"- ("'+0 ; 2/i—^- <^os. (ref+t)— ar. sin. (nt-\-i) ; hence may be in- dx dy ferred the values of — --^ — f^ , &c. &c. dx dx ^. COS. (»*+.) ={g}. cos.(»^+.)= {g}.cos.H«^+0-{g}. sin.(». + 0. COS. (.HO; ^. sin. («/+0= {f }•-•(«'+')= {g} • -• ^(''H^)+ {g} . sin. {nt + i). cos. (n« + s). .•. —~— cos. («<+0 + — tt"- sin. (nt-\-i) = -j-f n-x^.dt — PART L— BOOK 11. 97 CHAPTER III. First approximation of the celestial motions, or the theory of elliptic motion. 16. It has been already demonstrated in the first Chapter, that a body attracted to a fixed point, by a force which is inversely as the square of the distance, describes a conic section ; but in the relative motion of the body m about M, if this last body be considered at rest, we should transfer to m in an opposite direction, the action which m exercises on M ; therefore, in this relative motion, m is sollicited to« wards M by a force which is equal to the sum of the masses divided by the square of their distance, consequently the body m describes a conic section about M. But the importance of this subject in the theory of the system of the world, requires that it should be resumed under new points of view. For this purpose, let us consider the equations (K) of No. 15. If M+m be made = ji*, it is evident from No. 14, that if we only con- sider the reciprocal action of AI on m, Q is equal to — or to r fJ.U / o, the equations (K) will consequently become, dt= ^"^ h,u ,2 > PART I. BOOK II. 98 CELESTIAL MECHANICS, 0 =: —X +s. The area described by the projection of the radius vector, during dv the element of time dt, being equal to i. — =■ ;t the first of these IT equations indicates that this area is proportional to this element, and that consequently in a finite time, it is proportional to the time. By integrating the last equation we obtain s ■=. y. sin. (u— 8),t- * [f\ = _^_, /^l=_Z^3, <^m =0; therefore if these values ofi— >,^— ^,^-r-^ be substituted in the equations (K); tlie second of these equations becomes rf'zi , dQ s clQ d^u fi , u.s^ d'u + " - -d^l — T- ^7r= rxi +""-" + •- ' rfu* ^ du u ds — ^^dv^ "^ ^Vf+T*" k^(l+s^)^ dv' j^ u L_ 5 J and the third equation becomes • ' - h^[\+s'^y- dv" U.US uus d^s , + • — :r: — . dv. r"-. cos. '« r: tbe element of the area described in a given ti»e by the projection of the radius vector ; see page i. rf'5 d'^s.ds .,„,,,. ■ ds^ , . X + i = 0 ; ••• — ; h sds = 0, therefore by integrating —rrr + ** =: c, it dv^ dv' dv is evident that s = sin. v. or s = cos. ti, and that •.• s = a sin. v, or « = i. cos. v, and consequently s = a. sin. v-\-b. cos. v. will satisfy the given equation, and be its com- plete integral ; as it contains two independent arbitrary quantities. Now, a sin. v + 6. COS. V. may be reduced to the form y sin (v — 6), by assuming a — -y. cos. 6, b— — y. sin i, which gives a. sin. v + b. cos. ti = y. (sin. u. cos. 6 — cos. v. sin. f) — y. sin. (v'—e), and it may be shewn that y. sin. (d — 6), Ukewise satisfies this equation. It is also PART I.— BOOK 11. 99 y and 0 being two arbitrary quantities. Finally, the seconi equation gives by its integration « =T^7rT-5T •(^l+«' + ^' cos- (^'-^) \ = \/l+s* r .* e and isr being two new arbitrary quantities. By substituting in this o 2 evident, that s = a. sin, (d— 6) -[- o. cos. (u— ^) will satisfy the equation - + * z: 0, and may be used when convenient, but in this case a, h and i, must be selected in such a manner, that they may be reduced to two independent quantities. * In the equation -j-^ + m — ,,,,'" ,^3 , let P = '^ •% , and m = a. sin. (u — (l)-4-5. cos. (« — <) will be the complete integral of the equation -7-5- -|- «= 0; and a sin. (v — d) and 6. cos. (v — 6) will respectively satisfy the equation ——^ -f u — 0 ; now if the expression a. sin. [v — 6) + b. cos. (t) — S) be regarded as the integral of the differ- ential equation — ; \- v — P = 0 ; a and b must in this case be functions of the va- dv riables v, and as there is only one equation to verify by means of a and b, we can impose certain conditions on them whicli will facilitate their determination ; supposing them to be functions of v in the equation a = a. sin. (v — «) -f- *• cos. (u — 6), we shall have du =■ adv. COS. (y — S) — b. dv-sm. [v — 6) ■\- da. sin. (u — 6)-\- db. cos. (u — f); but as there are two quantities to be determined, and as the proposed question furnishes us with but one condition, we are at liberty to select the other condition ; for this pvir- pose let da. sin. (u — () + db. cos. (v — 6) = 0; then duz^ dv. {a. cos. (u — 6) — 5. sin. {v — 6)) ; and consequently, d'u=^ — dv'^. {a. sin. {v — f) -\-b. cos. (« — t)) -}- dv, da. cos. (v — I) — dv.db. sin. (y — 6) ; and this value of d'u being substituted in the equation — -— \- u — ,— — -3 gives, dv^ A^(l4-s^)T adv'' . (sin. (u — f) — sin. {v — 6)) +Wd^. (cos. (d — 6) — cos. (u — *) ) + da-dv. cos. (y — i) — db.dv. sin. (u — 6) — Pdv^ =0; ••• da.dv. cos*, (u — S) — db.dv. sin. (v — «). cos. (w— ') 100 CELESTIAL MECHANICS, expression for u, in place of s, its value in terms of v, and then sub- stituting this expression, in the equation dt — — — j- ; the integral of the resulting equation will give t in a function of v ; therefore we shall have V, u and s, in functions of the time. — P. COS. (v — 6). dv^ — 0; and if this equation be divided by dv, and then added to the equation da. sin*, (u — S) -\- db. sin. (u — 6). cos. (u — 6) =0, we shall have da z= P. COS. [v — 6). dv, of which the integral h a —a' -\-f P. cos. [v — 6). dv; in like manner if the same equations be respectively multiplied by cos. (v — 6), sin. (v — 6), we obtain by subtracting the second, divided by dv, from the first ; db= — P. sin. {v — 6) dv; and •.• hz=.b' — J P. sin. {v — i). dv. Therefore u = a. sin. (v — 6) -\- h. cos. {v — 6) = a'. sin. {v — 6) + sin. {v — 6) J' P. cos. (u — 6) dv. +6'. cos. (v — e) — cos. (v — i)./P. sin, (■a — i) dv ; a' and b' are the values of a and b when P — 0 ; P = ;,,. . — r;3 = (by substituting for s^ its value) -: —. ~3, therefore 6m.{v — I) /P. cos. (v — 6)dv = ^ -.-i -. f ^ : 3 , but ^ '-^ ^ ' A' •^(l-f-y\sin. "(u— e))'' cos, {v — 6) dv _ sin. (m — i) sin. (ti — 9) •^ /i^(l-|-y=.sin.i(t)— «)^ ^ A*(l-fy\sin.^-(u— 9)t ' *"^ ' /j'(H-y'.sin.^(u— (i)i COS. (u — e\dv y^.sin ''{t' — fl). cos. (ti — ^^.dv , , . ^ : — —1. — ; — ^ = by reducmg to a cotn- ^^l+y^sm. ^(«— 0)^ A'(l +y-. sin. '(■!;— «))i ' ^ co%.{v—6).dv K.sin. (p— «) mon denommator 77-- — — 3 ; consequently A cos, (t) — 6).dv _ ^ sin, ^{y — 6) •^(l + y^sin. ^(v — «;)! ""F"* (14-y^sin. ''0^— «)^* — cos. (u — S). f P. sin. (u — 6). dv = , , .\ /. sin. (t) — d). dv , sin. (n — 6). dv -^.(cos.i.-^).fj,^^--—---,^ , ,r.if.^^_^^—-—.^, _ —1 cos. {v—6) 1 COS. (f— 0 ~ (l+V^)* F(I+771iir>i::«)' ' 1+7'-" Ani+y^sin.a(x,_9))^ 1 sin, (v — 6). dv I ^ sin. (v—e). cos. ■'{v—e). dv l+y''h'{l-i-'y\sm.'{v—e))i i+ya-'i'- A"(l+y=, sin. =(v— «)) -^ : by reducing — —i fa'n-(t— ^) +y^ sin. (t,-0) (sin. ^(t^-Q+cos. ^(v-ll)).dv ^ + y'" A^{l+y^sin.*(«_(l)l PART I— BOOK IL loi I The calculus may be considerably simplified, by observing that the value of s indicates that the orbit exists entirely* in a plane of which y is the tangent of the inclination to a fixed plane, and of which 9 represents the longitude of the node, reckoned from the origin of the angle v. Consequently, if we refer the motion of m to this plane, we shall have s =0, and y = 0, which gives = — = -|^< 1-he. COS. (v — ;!r)>. This is the equation of an ellipse, in which the origin of the radii is at the focus : — rr- j-, is the semiaxis major, which we will repre- sent by a ; e is the ratio of the excentricity to the semiaxis major ; 1 sin. (u — (l).(l +y^). rfu sin. (v — 6).dv ' 1+yi* A'(l+ySsin. ^(u— «))4 A"(l-fy'.siu. ^(u— «))T ' ft. sin. {v — 6). fcoi. [v — 9). dv ft. cos. (v — (l)ysin. [v — S). dv /j»(l fy^ sin. '(v—6)f. h'(l+y\sm.''{v—6))i __ ftsiD.^(v—6) 1 ^ COS. "(u — e) ~ F(r+y^^'sin~HJ^— ^^ HV ' /j'.(H-y^ sin. i(v—6))^ ~ (sin. ''{v—S)+cos.^(v—e)-^y^.s\n.^v—e) _ (l +y ^. sin,^(v-«))^ (l+y)'./%»(l-|-y«.sin. «(d— 9)« " (l+y"). A« \i = f'- (i+y«)/^^ ' •'•'* = °'- sip.(v-e)+i'.cos.(.-^) + ^.-^\;;;^,;^.^ , and as e'. d'u cos. {v — w) satisfies the equation -yT + " = 0, we may write this function instead of a', sin. {v — 6) -\- 5'.(cos. (v — C), and as e is arbitrary we can assume it equal to / — — , • e, by means of which the expression for u will assume the form given in the text. * y is evidently equal to the tangent of latitude, when v — i = 90, and consequently it is in this case equal to the inclination of the orbit ; and as sin. (v — 6) = — = s. cotan- y gent of inclination ; the orbit described must be a plane, for this equation expresses the relation between the two sides, and invariable angle of a spherical triangle. 10^ CELESTIAL MECHANICS, dv finally, tb- is the longitude of the perihelium. The equation dt zz -j—, becomesj by substituting in place of i/, \/[A. (l+e. COS. (y — •B-))^ Let us expand the second member of this equation, into a series pro- ceeding according to the cosines of the angle v — ■a-, and of its multiples. For this purpose, we will commence by expanding >—- , ^_ into a eimilar series. By makiag X = 1 + ^1-e^' we shall have l+e.cos. (t;-..) v'r=?tl+ ^. c^"— ^^-^ l+^.c-^'-'^-'^-^J '■ * — = r = -7; ; r: — — n -, — ^r- , ••• a = — • ; hence h =1 u ^(l-|-e. COS. (u — •a)) 1+e. cos.(u— ar) jit{l — e») t By reducing the coefficient cX—=^ ^ \a. the second member of this equadon to the same denominator, it becomes equal to 1— ^" (u— ar)7^/_l _(u_sr)V— 1) •1— e=.(l + x» + ^(c^ '4- c but c — c =2 COS. (v — to), •.• this second member = PART I.— BOOK II. 103 e being the number of which the hyperbolical logarithm is equal to unity. By expanding the second member of this equation, into a series ; namely, the first term relatively to the powers of c ~^' * and the second term relatively to the powers c~^~'°)'^—y^ , and then substituting in place of the imaginary exponentials their expres- sions in sines and cosines ; we shall find 1 1 \+e. cos. (w — w) y/i ^' (1 — 2x. cos. (u — c3-)+2a^ cos. 2(y — =r) —2a'. cos. 3,(u— w) + &c.) ; By representing the second member of this equation by ?>, and making q — — , we shall iiave generally, 1 x» g , ; and from the equation a = ■ , we obtain VI— ««)(14A=' + A. COS.(t)— ar)) (l-j-v/l_e^) ~ \Tl/, n? ) . and 1 + A' = S J=i-; •.■ by substituting for !->.*, andl+A« we obtain 2(1— g^+y/l-e') ^ i 2.'/l_e"(l+^l— e").(l+e.cos.(u— w)) 1 + e. cos. (v—,,) * The expression of the first term gives the following series : the expansion of the second term gives making the factors of the same powers of a to coalesce in the two series, and observing . ^ i. ilv — a).</ — 1 , — i(v—'a)\/ — 1) i that A (c +c — A . COS. t{v -sr), we will obtain the value of l+c.cos.(v->)-' ""^'"^ '^ S'^^" '" t'le text. 104 CELESTIAL MECHANICS, e-'"-\dr^-^ ±______iJL (1-fe. cos.(w — •sr)) 1'2'3 m.dq^ in which rfg' is supposed to be constant, and the sign is + or — , ac- cording as m is even or odd. From this, it is easy to to infer, that if we make :!^ — (I gs^— I (1+e.cos. (t;— Tir)7 "~^ ^ (14-E^^\ COS. (v—sr) 4- -E^^^. cos, 2(i)— x^)+£(^). cos. 3(t)— TS-) + &c.) J we shall have, whatever may be the value of i. (l + \/l — e»>' the sign being +, if i is even, and — if i is odd ; therefore if n be ♦ Substituting — for e we obtain — 7; -. r =— '- = ip, :• ° ^r 1 "t^e. COS. (11 — st) y+cos. (d — ar) (9+COS. (u — w)) 9'" y + cos. (i; — w) ' " (^+003,(1) — w)' " 1 9 J ' iaiAd\\ — \=d- —■ -^dq= , % -, and di. 1^1 = [q) {g + cos.(v — T^y {ij-f-cos.(v — ar)-5 lyj =d.; ; rr .1- dg = ■ — ; -^ — • : hence generally we obtain d"' i — J- (y+COS. (e— sr))i • ^ (y+COS. (u— ar))+ ^ ' \ g j ^ ±. 1.2.3 w ± 1.2.3 me '^ 7W 4- 1 ?K 4- 1 (jr+C08. (v — ■a)) (1-fff. COS. (l) — to)) t Substituting — for e, in the value of ffl, we obtain— = .(1— 2>. cos. («—«;) + 2a'. ? ? •?»— 1 1 -2 f 0 ■) cos, 2(i>— «7)— 2a 3. COS. 3(r— s-)+ &c.) v 77-; ,,, =e .d.{ — Vz^\he ^ ' ' (1 +e. cos. V — a)Y \ 9 J dq PART I.— BOOK II. 105 supposed equal to a '•v /*, we shall have ) ^ (5 . COS. (v — u7)-\-E COS. Z(y — zs) + &c.) ; (1) (2) (3 ndt = dv. (l-h- E . COS. (v — 3-)-f £ . cos. 2(t' — ■o-) + E • and by integrating nt + t=v+E . sin. (f — zr^ + ^.E . sin. 2(t; — w) + ^£ . sin. 3(v — -sr) + &c. s being a constant arbitrary quantity. This expression for nt-\-i is very converging* when the orbits have a very small excentricity, such as the orbits of the planets and of the satellites ; and we can, by the PART I. BOOK II. p preceding series differenced with respect to q, and divided by e' ; the differential of the — 2 , 1 ^ i „ — 2 2 terra = e —2 V 1 ^^—21 2e. — r^x3'— — . ±2e (y+^y^_l)H-l = by simplifying and reducing to a common denominator, — °* ■ —, ,• , which becomes, by substituting — for o, , 2<r'(l + J VTH:?) ^ . . . , ^ 3 ^ - • , the expression given in the text. * (l—e'')^ occurs both in the numerator and also in the denominator of the value of n.o'^, as is evident from the value of dt given in page 101, compared with the preceding expres- sion ; when the excentricity of the orbit is inconsiderable, e which expresses the ratio of the excentricity to the semiaxis major will be very small, •.• the value of £(0, in which e' occurs as a factor will be very small, and perpetually less and less. 106 CELESTIAL MECHANICS, reversion of series, conclude the value of v in terms of /; we will effect this, in the subsequent N°'- When* the planet returns to the same point in its orbit, v is in- creased by the circumference v/hich is always represented by Stt ; nam- ing T the periodic time, we shall have This value of T may be easily deduced from the differential expression for dt, without recurring to series. In fact, let us resume the equation -^ dv r^.dv ^ . ,,■ rr r ^*-^^ at = , or at = — ; . from it, we obtam I := I — ; — ; h.u^ h ^ h rr^.dv is double the surface of the ellipse, and consequently it is equal to 27r. a*, tj \—e^ ; moreover, A* is equal to ^a. (1 — e') j thus we shall obtain the same expression for T, as has been given above. If the masses of the planets be neglected relatively to that of the sun, we have •/ ^ =z ^M; the value of /* is then the same for all the planets ; T is therefore proportional to a'^ , and consequently, the squares of the periodic times, are as the cubes of the greater axes of the orbits. It is evident, that the same law obtains in the motions of the satellites about their primary, their masses being neglected rela- tively to that of the primary. 17. The equations of the motion of two bodies, which attract each * When the j planet returns to the same point, the terms of this equation will be- come n(i4-r)-l-e = o-f-2x+£^ '.sin. ((r— sr)-}-2!r)+£^ '.sin. 2{(v—a)+2^) + &c. if this equation be taken from the equation nf-f- e = v+E^^\sm.{v—^)+E^^\ sin.2(u— ar)-f£;^^^8in.3(r— ro) + &c. the difference wUI he»7'=2T. PART I—BOOK II. 107 other in the inverse ratio of the squares of the distances, may be also integrated in the following manner : the equations (1), (2), (3), of No. 9, become, when we only consider the action of the two bodies M aud m, (O) (/A being equal to M + m). The integrals of these equations will give the three coordinates X, 1/, z, of the body m, referred to the centre of M, in a function of the time, and then by No. 9, we can obtain the coordinates ^, IT and y of the body M, referred to a fixed point, by means of the equations 0= ^'^ dt' + IJ..X 0- '^'y df- + + „3 Finally, we shall have the coordinates of m, referred to the same fixed point, by adding ^ to a;, n to i/, and y to z; by this means we shall obtain the relative motions of the bodies M and m, and also their absolute motion in space. Therefore every thing depends on the inte- gration of the differential equations (O). For this purpose, it may be observed, that if there is given be- ■ , 1 (1) (2) (3) (») tween the n variables x , x , x x , and the variable t, of which the difference is supposed to be constant, a number n of dif- ferential equations determined by the following : dx' , A.d a; B.d x- ' „ (s) de dt'-^ dt'-^ P 2 * In every equation of the same form as that in the text, if the a; ' , x^" ' ' "' 108 CELESTIAL MECIHANCS, in which we suppose that s is successively equal to 1, 2, 3, n; ^, 5, ...^ being functions of the variables a; , x , x , x , and of ^, symmeti-ical with respect to the variables x ,x , x » that is such, that they remain the same when any one of these variables is changed into the other, and vice versa, we can suppose J'^= a}V"--'+'^ + l.^'\J"-'+'^ + L^'\ /^ ^(2)^J2)^(„-.+l)^^(2)_^(.-H2) _^^^(2,^(„). („_i) (n—t) (n-i+l) An-i) {n—i + 2) Jn—i) (n) X :^a .X +y .x + /« x , a^ ', b h ; a , b , &c. being arbitrary quantities of which the number is equal to i(n — ?'). It is evident that these values satisfy the proposed system of differential equations : moreover, they reduce these equations, to i differential equations between the i variables X ' , X r . Iheir integrals will introduce «* new fjj i+3) In) X , « ) quantities satisfy this equation; then their sum will also satisfy the same equation, as will appear by substitution, and we are at liberty to assume ^(1)^^(1) /n_Hl)^^(i)_ ;«-+2)._.^(l),« In each of the values of ;^\.(2\ (n) X , there are i arbitrary quantities ; ••• in the sum of all the values of then — t quantities these are i.{n — i) arbitrary quantities. In the integration of a differential equa- tion of the i order, there are i arbitrary quantities introduced. .•. In the integration of i differential equations of the i order, there must be in all, i" arbitrary quantities. This theorem is evidently applicable to the differential equations (O) ; for these equations are symmetrical wirh respect to x, y, z, and remain the same, when any one of the va- ..,.,,. ,_ J (') («— «'+!) («— J+2) nables is changed mto another ; •.• as x, y, z, correspond to jr , ^ , x ■ Sec. in the theorem, we are at liberty to assume one of them z equal to the other two, mul- tiplied respectively by arbitrary quantities. PART I.— BOOK II. 109 arbitrary variables, which combined with the i.(?i — i) variables, already given, will constitute the arbitrary quantities, which would be pro- duced by the integration of the proposed difFei-ential equations. The application of this theorem, to the equations (O), gives 2=:ax-i- by, a and b being two arbitrary quantities. This equation is that of a plane passing through the origin of the coordinates ; consequently, the orbit of m exists entirely in the same plane. The equations (O) give but by differentiating twice successively, the^equation rdr = xdx+7/di/ -\-zdz, we obtain r.(Pr+Sdr.d^rzzx.dlx+y.d^y-\-z.d^z + 3.(dx. d*x+dy.d*y+dz.d*z). and consequently, By substituting in the second member of this equation, in place of d^x, d'y, d^z, their values determined by the equations (O'), and then, * rd'■r-\■dr*=xd^x + 1fd^'1/+zd''z■{.dx^■^dl/^+dz^, :• rd3r-\-Sdrd'r = xd^x+yd^j/+ zd^z-]-Sdx.d'x-{-3di/,d^i/ + 3dz.d'z, and multiplying by r' we obtain the expression in the text. no CELESTIAL MECHANICS, in place of (Px, d-y, d^z, their values given by the equations (O) ; we shall find '=4'--^\+^- » , ^..dj\ dt^ The comparison of this equation with the equations (O'), will give, in consequence of the theorem which has been announced above, ( dx dy d2 dr , . ■ , i ,. , I -r-j —jr, —j-f -J-, benig considered as correspondnig to the particu- dt dt dt dt • variables the time /;) lar variables x , x ,x , x , and r being supposed a function of dr zz X. dx+ y.dy ; A, y, being constant arbitrary quantities ; and by integrating, r — — +xa;+7^,t h^ — being a constant quantity. This equation combined with the fol- lowing : » From the equation (O') we obtain r'^.x. —j— = — Sr^.x. -^. dr — f^j^ilx, and by substituting for -7-^ , we have r^x.-— =3 ^ — dr—fudx ; .: the second member of the , „ (x=H-y^ + 2^ , (xdx\ydyi^zd£\ „,«»■'/,, . preceding equation = + 3^. \—Ll-L-.). dr — ^.-i 2_^^ 3 -^ {xdxSf-ydy -\-zdz), hence the second member is reduced to — /^..dr, wliich combined with the mem- ber at the right hand side, gives the expression in the text. f It is clear from an inspection of the equations (O') that the theorem already an- nounced, is applicable to them, and to this last equation, since any one of these variables dr dx may be changed into the other without affecting the constant quanttjes, ••• 'IT— ^- "IT ■^''■-dT' PART I.— BOOK II. Ill gives an equation of the second degree, between either x and y, x and z, or y and z, consequently the three projections of the curve described by m, about M, are lines of the second order, and therefore as all the points of this curve exist in the same plane, it is itself a line of the se- cond order, or a conic section. It is easy to prove from the nature of this species of curves, that when the radius vector r is expressed by a linear function of the coordinates x, y ; the origin of the coordinates must be* in the focus of the section. Now from the equation, rz= /* + A. a; + y.y, we can obtain, in consequence of the equations (O), ^■{-^} By multiplying this equation by dr, and then integrating, we sliall obtain rfr' ur° rK -^ — Qf^.r+ J^^ h'=0,f d being a constant arbitrary quantity. From which may be obtained ,^ rdr ' ' a fJL, this equation will give r in a function of t ; and as by what precedes, * It ig a distinguishing property of the foci of conic sections, that if their equation be expressed by means of polar coordinates, these coordinates will be linear, when the origin is at the focus. d'r d*x d^y r y.^ , y.y 1 < ^' ? + ^="--rfF + y^ =-^. ^--l-^^:^-^. Ir—y^' Multi. plying by dr, we obtain, -^-^ = — /». ~+k\-f; and by integrating -^ 112 CELESTIAL MECHANICS, X, y, z, are determined in functions of r J we shall have the coordi- nates of w?, in functions of the time. 18. We might arrive at these several equations, by the following method, which has this advantage, that it detennines the arbitrary- quantities in functions of the coordinates x, y, z, and of their first dif- ferences ; which will be extremely useful in what follows. Let us suppose that V = constant, is an integral of the first order of (ijc {In dz the equations (O), F being a function of .t, y, z, —7- , — , — : Let X', yy z , represent these three last quantities, and then the equation V = constant, will give by its differentiation. dz_ dt ^_ §dV\ dx <dV\ dy CdV\ Xd^S' dt'^Xdyj' dt'^XdzS- CdV^ dx idV\ dy'§dV-k dz' , but the equations (O) give dx _ fj.x dy' _ \^ ^_i jt-- 1t~ 1^' ~di~ '^'dt- r'' consequently, we have the following identical equation, of partial dif- ferences, HSdV) , UV} , SdV)\ ^^^ '■W\+^-U'\-''-id7U' It is manifest, that every function of x, y, z, x', y\ z', which, sub- stituted in place of (F) in this equation, renders it identically nothing, • As F is in an immediate function of the six variables, x, y, s, x, 1/, z', its differential coefficient with respect to another variable t, must be equal to the several differential coef- ficients of V, considered as a function of x, y, z, x', y, z, multiplied respectively, into the differential coefficients of these variables, considered as a functions of i. PART I.— BOOK II. 113 becomes, when it is put equal to a constant arbitrary quantity, an in- tegral of the first order of the equations (O), Let us suppose V= U + U'^-U" + &c. U being a function of the three variables x, y, z; U being a function of the six variables x, y, z, x', y', z', but of the first order i-elatively to a/, y', sf ; U" being a function of the same variables, and of the second order relatively to x', y\ z', and so of the rest. Substituting this value in the equation (I), and comparing separately, first, the terms in which x, y, 3', does not occur ; secondly, those which involve the first power of these variables ; thirdly, those which contain their squares, and their products, and so on of the rest ; we shall have ( ^dU'i^ S'^U"-)^ S'^U'l-. &c. The integral of the first of these equations is, as we know by the theory of equations of partial differences. hao PART I. BOOK II. 114 CELESTIAL MECHANICS, TJi •=. func. {xy' — yod, xz' — zx, yz' — zj/, x, y, s.)* As the value of V must be linear with respect to x', y', z', we shall suppose it of the following form : U' = A.(xy'—yx)+B,(^xz' — zx') + C.(yz' — zy') j A, B, C, being constant arbitrary quantities. Let the value of V be continued as far as the term U", so that U'", V"", Sec. may vanish ; the third of the equations (F) will become The preceding value of U' satisfies also this equation. The fourth of the equations (I') becomes The integral of which equation, is JJ" — funct. {xy — 3/a;', xz' — jsa/, ysi — zyf, x', y', z')A This function ought to satisfy the second of the equations (!'), and • For the integration of this equation see Euler Integral Calculus, tome 3, chapter 3, No. , and Lacroix Traits Complete, Tom. 2, No. 634. I P being the derivative function of V, —ry = — (i'+z). F', — - = (i-s). F*; —-^, = (X ■Vy)-F;:-x.-^+y.-^-\rZ.-^= (-«.(y+«) + y.[x-»)-\.z.{x +y)).F' = 0; -j-=(y'+s').F'; -7— =(z'— x). i^'; — - = — (x +/). F'; .-.x. + y. ^ +z'. ~—= (x'.(/+z')+ y'-(z'-^') — «.(x'+y).r'= O ; Multiplying the (dU dx , dU dy , dU dx-) , first member by dt, and substitutmg we obtam j -- — • -jt.+ -j— . -^ + ~t~' "jT i • " = dV. PART I.— BOOK ir. lis the first member of this equation multiplied by dt, is evidently equal to dU ; therefore the second member must be an exact differential of a function of ^, y, z. But it is evident that we can satisfy at once this condition, the nature of the function U", and the supposition that this function is of the second order in z', y', x ; by making U" = {Dy'—Ex'). (xy'—yx') + {Dz'—Fx'). (xz'—zx) -1- (Ez'~Fy'). (j/z'—zy') +G.(x"-ty'°-+z'')i D, E, F, G, being constant arbitrary quantities ; and then r being equal to ^/x^+y'+z^, we have U =— -^. (D,T+Ey+Fz+2G) ;• Q 2 dU • "^l- — =_ D^y + «-) +£.(2^x'-xy)+J'.(2xz--xi')-}-2G*', dU" -^ = D^yx-yx') — £.(xx'+«2') + F.(2ry— yz') +20/, -^ = D.{2z'x _rx )+£.(2y2'-;:y)-P.{w'+yy)+2Gs'. dU" dV dU'^l da/ '*'^' rfy +^- rfz' 3- — Z).((yxy+ zxs/) + E. ( 2yx.x'— X 'yO + F.{2zxJ—i^z' + 2Gxx') -^ + Z).((2xvy-y^x')-£.(xyx' +zys:')+F.(2zy.y— y'r')+2Gyy) J5L + D.{(2xzz'—z'x')+E.(yzi'—z'7/')—F.(xzxi- yzy") + 202/) -^, = by concinnating and omitting those terms which destroy each other, ( — Z).(y^-j-z') ^-E\x-'Jr~')-i/—F.[x'-^z^y^D.{xy)y'-irD.{xzy-{-E.{yx)x'^E{yzy^F.(zx)x'+F.{zy) y + 2G.(xx' 4- j^y'+sz')) -^ = (by observing that y' 4-i'=r^— x^ ; x'-|-z»=>-»_y^ 4c.) the value of 17, differenced with respect to x, y, 2, successively, for 1^ = - f • -0+73 -^^^ + Ti (£i/*+^«+2Gx) = _ -^. (2).(y'+z»)-%x - 116 CELESTIAL MECHANICS, consequently we can obtain, by this means, the values of U, U', U" ; and the equation V zz constant, will become const. = — 4- {Dx+EyJ^Fz+2G)^{A-\-Di/—Ex').{xy'—ya:')* ■ + {B + Dz'—Fs').(xz'—za:') -i- {C-\-Ez—F>j').{yz'—zy') This equation satisfies the equation (I), and consequently the dif- D.xy-F.y.-1Gii), ^ = - ■^- F.^. ^ . Fz^+ ^ (Dxz+Eyz+2Gz) = - fr- { J (.T*4-y*) — -Dx2 — Eyz — 2Gz), ••• if these equations be multiplied by :c', y', J , respec- tively, the sum of the terms at the left hand side will be equal to d\], and the sum of those on the right hand, will coincide with those already given. * This equation evidently satisfies the equation (I), for ir = _il. D. + — . (,D.v'+Exy+Fxz4-2Gjc)+Ay+Dy"—Ex'y'+Bz'-^Dz-—Fx'z' dx r r^ ^=—^,E + -^.(Ey''+Dxy-\-Fyzj-2Gy)—Aj^—Di/'se'^Ex''-iCz' + Ez'''—F,/:^, dy r r^ '^Z.= .-fl.F+—. (Fz^ + Dxz4-Eyz-{-2Gz)—Bx'—Dz'x' + Fx"—Cy'—Ez'y'+Fy'^. dz r r^ , dV , , dV , , dV dx dy dz _JLm)f-\-z^)—Exy—Fxz—2Gx).x'.+Ay'JJrDy'^-v'—Ex''-y'-\-Bzx^Dz'^x'-F£'z', _ iL. (£(xH2" )-Dxy-Fyz—'iGy),/-Ay'x'-Dy'^x+Ei^^y'-\- Cz'y'^Ez'iy'-Fy^'z' , — ^. {F{x^-^y^)-Bxz -Eyz->f2Gzy-Bx'z'-Dz''x>J[. Fx'^z'—Cy'z'—Ez''y'+Fy"z: = by obliterating the quantities which destroy each other _fL.(Dh/''+z')-Exy-Fxz-2Gx)x'J^E{(x^^z')-Dxy-Fyz-2Gy)y'^F{x^^y')-Dxz-Eyz -2Gz)z'; '^=—E{xy'—yz')—y{A + Dy' -Ex' ]—F{xz'-zx')-z{B+Dz'-Fx-)+2Gx, PART I.— BOOK II. 117 ferential equations (O), whatever may be the arbitrary quantities J, B, C, Z),. ^. F, G. Supposing them all to vanish first , with the exception of A ; 2dly, with the exception of B ; Sdly, with the „ „ . , . . dx dt/ dz . , c > ' , exception of C, &c., and restoruig — — , -^ , -r:, m place oix,y, z', dt dt dt we shall obtain the integrals c = xdy—1/dx ^_ xdz — zdx _ „ _ydz--zdy__ \ dt c'zz dt ; C" - dt n-f.^S'^ (dl±dz-)l ydxj.dx zdzxlx .. "-•/+'^-\7 -Of )■+ de ^ df ' f* {da^^dz^) \ xdx.dy zdz.dy \ S^ dl^ "^ df ' I a~f"j.. S ^ {dx^^df) ) xdx.dz ydy.dz^ _ "-■/ +^-|7 2f > dt^ ^ de ' a r dt- (P) / c, c\ c'', f,f',f", and a being constant arbitrary quantities. '^-L = B{xj/-t,jf)-^x{A+D^--E=if)-F{y^-Z!/)-z{C+E^-Fy')+'2Gy', ^ = D{xJ-zxf)-\rx(B^D^—Fx-)J^E{y:^-Z!/)+y{C^Ez'~Fy')-^^Gz', ' Multiplying these three equations by x, y, z, respectively, and observing that those terms, of which one factor is the product of two of the coordinates, x, y, z, destroy each other, we obtain, by concinnating —r-;X-{ — tt- V + -r-r- z^ — £(x'+z^)v' — i)(y^+3*U'' — ■' ^ dx dy' -^ dz ^ ;,y w , Fiy'' +x''y+E{yx)-\-Fxz+2Gi)x-^(Dxy+Fzy+2Gy)iJ+{pxz+Eyz-^2Gz)z', and It dV this expression, when multiplied by -^ is identical with the preceding valueofj:'-^ \- , dV , dV .V--T- + ^ dy dz * Supposing all the constant quantities but A to vanish, the preceding equation be- comes const. =^A{xy' — ^x') ; supposing them all except D to vanish, we shall have const.= 118 CELESTIAL MECHANICS, The differential equations (O) can only have six* distinct integrals of the first order, by means of which, if the differences dx, dy, dz, be eli- minated, we shall obtain the three variables %, y, z, in functions of the time t; therefore one at least of the seven preceding integrals should occur in the six others. We may perceive even, a priori, that two of these integrals must occur in the five remaining. In fact, as the sole element of the time, occurs in these integrals ; they are not sufficient to determine the variables x, y, z, in functions of the time, and conse- quently tiiey are inadequate to the complete determination of the mo- tion of m about M. We proceed to examine how it happens that these integrals are only equivalent to five distinct integrals. zdii^^v d '*' If we multiply the fourth of the equations (P) by — , » ^^^ then add it to the fifth, multiplied by — ; we shall obtain A- r i^dy—ydz) {xdz-zdx) {xdy—ydx) ^-J' It ^^ ' dt + ^' dt C|iA (dx^ + dy'^) } (xdy — ydx) f xdx.dz ydy.dz\ t (T If i"** di \~~dF"^~~df~S' yy.J— zzV which will be equal to the fourth of the equations (P), by substituting for x', y, ^, their values. Supposing G to be the only constant arbitrary quantity, we ob- lu. , , . const. u, , , . . tain, const.= G { — + (^"+i/^-\-'^-))\ ■•' makmg — - — = -i- , and substituting r for x'. y, j/, we obtain the expression given in the text. * As the differential equations (O) are of the second order, and since the complete integration of each equation furnishestwo constant arbitrary quantities, the entire number cannot exceed six. f Performing this multiplication and addition, we obtain ft. (xzdy—xydz)—xzdy^—xzdy.dz''+xydy-.dz-\-xydz- , zydy^ .dx-\-zldxdydz PART I.—BOOK 11. 11 T, 1 ^v x- • 1 r z'hi~—udx xdz — zdx ydz — zdy , . By substituting m place of — - — -^ — » ; 1- — their ' 6 f ^; dt dt values, which have been determined by the three first of the equations (P), we shall have 0 —f^'—f^" -L o. Sit _ C^ljh^l X . ^^f^^dz ydy.dz c '^ Ir I dt' S)'^ ~dF~'^~^t''~' This equation coincides with the sixth of the integrals (P), by making ./' = LE^ZiJL, or 0=fc"—f(f+f'c. Thus the sixth of tlie integrals (P), results from the five preceding, and the six arbitrary quantities c, d, c'', f, J", f", are connected together by the preceding equation. If we take the squares of the values o? f, f, f", which are deter- mined by the equations (P), and then add them together, we shall obtain — y}.dx.dy.dz — yz.d*z.dx fi j t f^ (xy.dz — t/zdx) — yxAx^dz — yxdz^ yz.dx^ + yz.dz'.dx x^.dx.dy.dz-\-x.zJz'.dy xz.dx'.dy z'.dx.dif.dz + ^3 + -dT^ df^ d^ =by mak- ing factors to coalesce-/c'^i/'c'+..^ iffc^ _ z. ±^ (J^l+^ ^ ,. fd_y ^ -^ ^ ^ r dt dt dt" ~ dt {dz^—dx'^)xy.dz(dy^ + dz-) xy.dz (dx^+dz^) _ y.dx {dy^~dz^) y.dx ^^ '^'Yt d? dT dt' + ^" dt df" "^""ir * {dz^+dx') , , ,^dx.dydz , , . dxdy.dz „ . . , ., -^: ^^ i- + («'— ^') — ^ij— + (x'— «'). j^ — = after all reductions, and obli- , . , , , , , (zdy — ydz) (xdz — zdx) su terating quantities which destroy each other, j — r/ • j, 1- — ^. (xdy — ydx) (xdy—ydx) {dx^ + dy"^ ) xdy ^ydy.dz xdxJzl ydx ~^t ^"^^ 1? ^~dr'i de + dt'' s~~dr' € xdx.dz , ydy.dz) ),.,., ... < — — — V , 1 i which IS the expression in the text. 120 CELESTIAL MECHANICS, /~K=(^- 1 a? S~lTtS \' I d? T> in \vhiclW= is, for the sake of abridging, put equal to /'' + /* 4-/"% but if we take the square of(, tlie values of c, c, c", which are given by the same equations, and then add them together, we shall have, by making c + c' +c = n ; *•>'"— ~7r- + ^ ' dt^ r ' dt' rfdy'^.dx^+z''dz-.dx^ 2yz.di/.dz.dx^ l^x (ydy.dx-\-zdz.dx) ^ (dy*+dz^) + '' di^ *■ dt* + r dt' ■^' ^i" (ydy.dx^zdz.dx) .„_ j^V, , (</a-*+2(fo'.&»+tfe*) _ 2^,y'- (rfx'+</z') dt' '-^ "" r" 4-^' rfi* r c^^* x'-dx'^.dy- -\-z''dz-.dy'' '2,xz.dx.dz.dy'^ If/.y (xdx.dy-^-zdz.dy) dx' -\-dz^ + '~~dF '' ' dF ^~ di^ ^' W^ (xdx.dy+zdz.dy) _ f^ _, (dx*+2dx\dy\+dy*) _ '^ ^, (dx*+di/^) d? '-^ - r^'^~- dt* r " ' dt^ x^dx\dz''+y''dy\dz'' , Qxy.dx.dy.dz" 2ft (xdx.dz+ydy.dz) ^ (/<+ ~ (/«■♦ ^ r di- (dx^+dy^ ^ (^dx.dz+ydy.dz) ^ . , ^^ obtam/'+/'^+/*^-^^= c/i' eft' * '-^ 3F^^ ^ y'' d0 '^^' di* " r '^ ■ ^<-^ r dt'- r ' dt^ "^ dt' dt* "^ dt'' dt' "^ rf/^ dt^ dy.dz dx- 2xz.dx.dz dy- 2yz.dy.dt dx'^ 2k ( rfw.c?^ rfx.c/z , dx.dy , du.dz , £/:r.tfe , dy.dx ■) PART I.— BOOK II. 121 X dl' J X dt S " ' consequently, the preceding equation will become, PART I. BOOK II. R C dx.du dx.dz 7 dt/'^ -\-dz^ C dy.dx du.dz 1 ,2, 2 , ,^ {dx*+d,/* 4-dz* + 2dx-.di/' i-2dx'-dz'-\-2dy'>.dz') {x +y +z% ■■ ~ X . (dx^+^dx^dy^-^-^dx^'.dz^) , (rfi/*4-2^■^'■rfy'+2%^f/^°) _^ (dz*i-2dz^.dx'-^2dz--d,,') dt* ^ ' dt^ " ' df- dx-dy ,^ dxdz , ^ dii.dz idx'^A-dy^-X-dz') 2/* , , , ((/j'+rfy''4-&') 2^ (■T^f/c''.4-,yVv'+zV/;:-42-r.y-c?-r-"'.y) (2xz.dx.dz-\-'2yz.dy.dz ) dp •" r M* "^ dt'' , , (du\dx''+d!/\dz') '(dx\dy'+dx\dz') , , {dz'du^+dz\dx') „ ^ literating tlie quantities which destroy each other, and observing that r''dr^ =x'</x*+ y*di/^+z'dz'+2xi/.dx.di/ + 2xz.dx.dz+2yz .dy.dz , dx''Xdij*Jf.dz^4-2dx'^.df-\2dx''.dz^A-'idui.dz'') , , , r'. ^i-^l-l. X ^|_I -1 '~(—x\dx*—y-dy*—z-dz'' f 2xif.dx.dy 2xz.dz.dy 2yz.dy.dz'' \ rdx^ + dy-^dz' l 2fc 2^ t (it-' dt^ dt^ l*\ rf<^ J— —•»■+— • -! -— • V , which may be evidently reduced to the expression in the text. * Squaring these equations and then adding tliem together, gives ^ {dy^+dz^ ^ {dx^-\-dz') __ {dx^+dy^ _ 2xy.dy(dx 2xz.dx.dt *■ dt' '^^'' dt"- "^"'' dt^ dt' dt" 2yz.dy.dz _ (dx^- +dy'-i-dz^) x^.dx' y'dy- z'^.dz^ ly.dz _ J {dx-+di/^+dz-) ( rdr \' ~' *■ at~ i"^J • dt- \ -ry -r I- ^^, ^^, ^^j j^ ^xyAx.dy 2xz,dxdz 2yzdy.d: dt* dt" dt' 122 CELESTIAL MECHANICS, ° = d? 7+""F"* The comparison of this equation, with the last of the equations (P), will give the following equation of condition h' a' Therefore it follows, that the last of the equations (P), occurs in the six first, which are themselves only equivalent to five distinct integrals, the seven arbitrary quantities c, c', c", f, f, f", and a being connected by the two preceding equations of condition. From hence it results, that we shall obtain the most general expression for V, which satisfies the equation (I), by assuming for this expression, an arbitrary function of the values of c, c', d', J\ andy, which are determined by the five first of the equations (P). 19. Although these integrals are inadequate to the determination of ic, y, z, in functions of the time, they .nevertheless determine the species of the curve described by m, about M. In fact, if we mul- tiply the first of the equations (P), by z, the second by — y, and the third by x, we shall obtain, by their addition, 0 — cz — c'y ■[■c"j:,* which is the equation of a plane, of which the position depends on the constant quantities c, c', c". If we multiply the fourth of the equations (P) by a;; the fifth by y, and the sixth by z, we shall obtain * Performing this multiplication the members at the right hand side of the equation will disappear, for they become _ xzdy — yz.dx — xy.dz + gi/.dx -\-yx.di. — zx.dy _ cz-<iy\-<;x —^ —j^ —2^ 0. PART I— BOOK II. 1S3 but by the preceding number we have, '■• W- IF- ' consequently, 0=!.r — fr+fv+fi/+f"z. This equation, combined with the following, namely, 0 = d'a: — c'j/ + cz ; r" zz x^ + t/" + z* ; gives the equation of conic sections, the origin of r being at the focus. From this it follows,* that the planets and the comets describe very nearly conic sections about the sun, this star existing in one of the foci, aud these stars move in such a manner, that the areas described by the radii vectores, increase proportionally to the time. In fact, if dv re- r2 * Performing this multiplication, and then adding the products together, we obtain rfu' , dz' dx.dy , „ di dz du dz (d£jd£A^ , dr^ dt" + '■ • dt^ ' From the first of these equations we obtain ^/'•/"•yt and by means of the equation 0 = c"j: — c'y-\.cz, and r* = x- +y^-Uz^ , we can eliminate, 2^ and z, and then substituting for r* its value, we arrive at an equation of the second degree between y and x, by similar process we obtain equations of the se- cond degree between x and z, y and z, from which it follows that the curve described is a conic section ; and as the value of r is given in a linear function of the coordinates X, y, s, the origin must be at the focus. 12* CELESTIAL MECHANICS, represents the indefinitely small angle, intercepted between the radii r and r + dr, we shall have dx* + cIt/- -\-dz^ = r Vt)« + dr^;* the equation ^ (dx'^ + dij^ + dz^) r*dv* _ , , ' ■ jr* di^ - ' will consequently become, r*dv'^ = h^dt^ ; therefore hdt dv = r« • From this it appears that the elementary area ^rdv, described by the radius vector r, is proportional to the element of time d/, consequently the area described in a finite time, is proportional to this time. It also appears, that the angular motion of m about 3J, is at each point of the orbit, inversely proportional to the square of the radius vector; and as we can, without sensible error, assume very short intervals of time, for the indefinitely small moments ; by means of the preceding • The differential of the curve z= ds = \/ dx+dj/'+dz^ = the hypothenuse of a right angle triangle, of which one side = dr, and the other side about the right angle ~rdv, :• dx' +di/''+dz' = ofs^ = dr'+r^.dv^- As h varies as the square root of the parameter, it follows that tlie angular velocity varies as the square root of the synchronous areas divided by the square of the dis- dt tance, see page 10 ; hence the angular velocity in a conic section is to that in a circle at the same distance r, as A : : v r ; ••• they are equal at the extremity of the focal or- dinate ; substituting for h its value g'raVl— e' dv_ ^.^^ Q^a.'x/l—e- . .^^ T ' dt T.r^ body describes a circle at the unity of distance in a time equal to T, then the angular velocity in the circle — -:^= the mean angular velocity in the ellipse, consequently, when the angular velocity in the ellipse is equal to the mean angular velocity, we havt gy _ 27ra . —e ^ ^^^ ^ _ ^^^j _ ^^^i^ _ ^ ^g^jj proportional between the semiaxes ; in this position the equation of the centre is a maximum. PART I.— BOOK II. 125 equation, we can obtain the horary motions of the planets and comets in different parts of their orbits. The elements of the conic section described by m, are the constant arbitrary quantities of its motion ; they are consequently functions of the preceding arbitrary quantities c, c\ c, J, f, f', and — ; we now proceed to determine these functions. Let fl represent the angle which the intersection of the plane of the orbit with the plane of a; and of y, constitutes with the axis of x, which intersection is termed the line of the nodes ; let ip be the mutual inclination of these two planes. If x and y represent the coordinates of ni, referred to the iine of the nodes, as axis of the abscissce ; we shall have x' = X. COS. 9 4-?/. sin. 9 ; y' = y. COS. fi — X, sin. 6. We have also z = y'. tan. (p ; consequeutly we shall have z = y, COS. 9. tan. (p — x. sin. 6. tan. p. The comparison of this equation with the following, 0 =. c"x • — dy-^cz'y will give d — c. COS. 9. tan. <p ; c" = c. sin. 9. tan. ip ;* from which may be obtained r , . . c' • A companson of these equations, gives y. cos. t, tan. ^ — x. sin. i. tan. ^= — , c" d I — . X •.• — = COS. 4. tan. ip ; c c See page 3, and page 34 of 1st Book. -'/« c" c* c" c' ^ +c" y — . x •.• — = COS. «. tan. ip; — = sin. i. tang. <p, •.• ^j = tang. *f. — C C C G 126 CELESTIAL MECIHANCS, tan. 6 = — y ; c tan. , zz ^/f+Zl. c By means of the preceding equations, the positions of the nodes, and the inclination of the orbit are determined in functions of the constant arbitrary quantities, c, c', c". At the perihelium, we have rdr = 0 ; or xdx + ydy + zdz -=.0 ; let therefore X, Y, Z, represent the coordinates of the planet at this point ; and from the fourth and fifth of the equations (P), of the preceding No. may be obtained. But if we name / the longitude of the projection of the perihelium, on the plane of x and of j/, this longitude being reckoned from the axis of X, we have Y consequently, X ~ *'^"* ^ ' f tang. I =-f> this equation determines the position of the axis major of the conic section. * Substituting — xdx ior ydy + adz, and —ydy for xdx -{-zdz in the two last terms of the second member of this equation, and they will become .-. multiplying the first by Y, and the second by X, and then subtracting, we obtain the expression given in the text. / PART I.— BOOK II. 127 If by means of the last of the equations (r), -tjz be eli- df minated from the equation r . — ,:^ ro-= « > we shall '■ at' dr obtain ~di ^.r-J^-l^^^h- but dr vanishes at the extremities of the greater axis ; therefore at these points we have, The sum of the two values of r in this equation, is the axis major of the conic section, and their difference is equal to twice the excen- tricity ; thus, a is the semiaxis* major of the orbit, or the mean dis- tance of m from M ; and v 1— is the ratio of the excentricity to the semi-axis major. Let e represent this ratio j and by the pre- * The coefficient of r with its sign changed is the sum of the two values of r, and their difference is equal to twice the radical, and •.* = to 2 a. y 1 , and V-'^ \/ 1 is the ratio of the excentricity to a ; \/ »« — -^-— =s fte fi. \ ft • r = ;«'e* = i' ; c?r = ae. sin. udu, •.• rdr = a\e. sm. udu.{l — e cos. u\ 2r = a.((2 — 2e. cos. a)— ( 1 -}- e* . cos. * « — 2e. cos. «)) := a.( 1 — e » , cos. *u), and V 2r a.(l — e') = ac^.(l — cos. *u) = ae*. sin. *«, and therefore rdr , , a'-.e. sin. u.[\ — e cos. m) du a^ ; _ __ - {~dt\ = = — -. V^.\j2r-rl.-.a.{X-e^). •^^/«^sin.^« V'^ (1 — e COS. u)du. 128 CELESTIAL MECHANICS, ceding number, we have a ~ k' ' therefore f^e ~ I. Thus, we can know all the elements which deter- mine the nature of the conic section, and its position in space. 20. The three finite equations found in the preceding number, be- tween ,r, 7/, z, and r, give x, y, z, in functions of r ; thus, in order to determine these coordinates in a function of the time, it is sufficient to have the radius vector r, in a similar function, which requires a new integration. For this purpose, let us resume the equation a ar by the preceding number, we have, therefore we shall obtain rdr dt = \/ju. \jlr~- a.(l— e°) In order to integrate this equation, let r ■=. a.{\ — e. cos. w), we shall have at = — -pz^. (I — e cos. U), from which may be obtained by integrating, t -^ T =■ —p.' (w — e sin. u) ; (S) T being a constant arbitrary quantity. This equation determiuM u. PART I.— BOOK II. 129 and consequently /• in a function of ;; and as x, y, z are determined in functions of r ; we shall obtain the values of these coordinates, for any instant whatever. We have thus completely integrated the differential equations (O) of No. 17 ; this integration introduces the six arbitrary quantities a, e, I, 6, <p, and T: the two first depend on the nature of the orbit ; the three following depend on its position in space ; and the last is relative to the position of the body m, at a determined pe- riod, or, what comes to the same thing, it depends on the instant of its transit through the perihelium. Let us refer the coordinates of the body tn, to other coordinates which are more convenient for the usages of astronomy, and for this purpose, let v represent the angle which the radius vector r makes with the greater axis, reckoning from the perihelium ; the equation of the ellipse will be a.n—e') r — ^^ — • i+e. cos. V The equation r — a.{\ — ecos. u), of the preceding number, indicates that u vanishes at the perihelium, so that this point is the origin of the two angles u and v ; it is easy to shew, that the angle u is formed by the greater axis of the orbit, and by the radius drawn from its centre, to the point where the circumference described on the greater axis as diameter, meets the ordinate drawn from the body m, perpendicular to the greater axis. This angle is termed the excentric anomaly, and the angle v is the true anomaly. A comparison of the two values of r gives 1 — e. cos. u = 1+e. cos. V from which may be obtained PART 1. BOOK 11. S ISO CELESTIAL MECHANICS, tang.i .r = v/l±£. tang, i ii.* 1—e If the origin of the time t be fixed at the very moment of the passage, through the perihelium, T will vanish ; and by making, in order to abridge, — f^ ~ )^ yfQ g\^^\\ have, ntzzii — e. sin. u. 1^ a"- By collecting together the equations of the motion of m, about M, we shall have nt := u — e. sin. ti, \ r — fi.fl — e. COS. u) f tan. 4 u = V -^- tan. i u. \ 1—e - J the angle nt being what is termed the mea7i a?iomali/. The first of these equations determines u in a function of the time /, and the two remaining equations will give r and v, when tc shall be determined. The equation between u and v is transcendental, and can only be re- solved by approximation. Fortunately, from the circumstances of the celestial motions, the approximation is very rapid. In fact, the or- bits of the celestial bodies are either almost circular, or extremely ex- sin. a + sin. i {"-\-i) , , , sin. a a * 7— — tan. — - — , let 6 :r 0, and , = tan. -- , i. e. COS. a-}- COS. 6 2 l-}-cos. a 2 v^'-^»^- "-" ^ ^^Lh-A«i-«_ = tan. 4- ; now .. cos. u =e i£+^hA, and cos. v = J + COS. a v/i+cos.a ^ 1 + c.cos.^ 1 — e. cos. M ' " '2 IL_</l— cos. v V^'^T _ cos. u cos. M — e tan. \/l-4- cos. u A / , . — e+ COS. m V 1—e COS. u y/l -f c— e. COS. tt— coslt ■/(l-f-e). (1— COS ;<) V^l-f-e « ■\/l— e— e. COS. H+cos. u ~ V(i—e).{l-\-cos. u) s/]ZIe ' 2 • PART I.— BOOK II. 13 i centric, and in these two cases, we can determine tt in terras of t, by very convergent fonnulEe, which we proceed to develope. We shall give for this purpose, some general theorems on the reduction of functions into series, which will be extremely useful in the sequel. 21. Let u be any function of a, which it is required to expand into a series proceeding according to the powers «. ; this series being supposed to be represented by u, q, qo, &c., being quantities independent of a ; it is evident that ii is what u becomes, when a, is supposed to be equal to cypher, and that, whatever be the value of n, {^}= ^'^-^ n.g„+2.3 («+)i.«y^_j^j-j- &c. the difference j—z — f, being taken on the hypothesis, that in u every thing is made to vary which ought to vary with «. Consequently, if we suppose that after the differentiations, a=:0, in the expression of s — r- Ldx. J' we shall have 9n = Xdl'^y 1.2.3 n If M is a function of the two quantities a and «', and it is proposed to expand it into a series, proceeding according to the powers and pro- ducts of a. and a.' J this series being represented by U ZZ. U + a.yi,o-|-a".5'2,o4- &C. 4- a'-.yo,o+ &C. s 2 132 CELESTIAL MECHANICS, the coefficient q„,„, of the product «".«'", will be in like manner equal to j- /+•» I Uo^\dcc"'' j 1.2.3 W.I. 2.3..,. ...v! ' a and a' being supposed to vanish after the differentiations. In general, if u is a function of a, a', a!'. Sec, and if it is proposed to expand u into a series, ranged according to the powers and products of «, a', cc". Sec, the term of this series, of which the factor is the product a.". a"" .ex.""'' will be aVa"'.a''.a" q^^ ,^, ,,^„ we shall have 1n!ri."kz. i ,» ,n , „n" „ f I. f/a .rta .fla &C. J 1 .2.3 nA.'i.Z n. 1.1.2, ii . &c. ' provided ot., a.', a,". Sec, are supposed to vanish after the differen- tiations. Let us now suppose that u is a function of a, a,', a.'', &c, and of the variables t, f, t", &c. ; if by the nature of this function, or by an equation of partial differences which represents it, we have obtained i^da". dcx,^ . &C. J in a function of u and of its differences, taken with respect to /, t, &c. ; F representing this function, when u is changed into u, u being what u becomes when «, «', &c. vanish, it is manifest that we shall obtain q„,n,, &c. by dividing F by the product 1.2.S...M. 1.2.S...«', &c. ; therefore we shall obtain the law of the series according to which u is expanded. In the next place, let u be equal to any function of t-\-a, f-\-x\ f ■{■»', &c., which we will represent by <f(J-^a, t'-\-a!, t"-\-oi"), in this PART I— BOOK II. 133 case the ?n'* difference of u, taken with respect to a, and divided by (/as"*, is evidently equal to this same difference, taken with respect to t, and divided by dt . The same equality obtains between the differences taken relatively to a.' and t', or relatively to a." and f, &c. j hence it follows, that in general, we have I a u I— J d ^ ( \dx.da.'^ .c?»"" . &c. J t di'.dt .dV'' J If in the second member of this equation, u be changed into u, that is, into>(^, f, f, &c.) } we shall have, by what precedes, ,. _5 /+"'-^""^'^".Ku'.r.&c.) I ^"■'-•""' •" 1 1.2.3... «. 1.2.3...n'. 1.2.3...n".&c.i If M is a function of t and a, only, we shall have ^'' ~ 1.2.3. .M.dr' therefore K^+.)_,(/)+— ^^+_-^^^+^— ^.^-^ + &c. 0) Let us in the next place suppose that u, instead of being given im- mediately in a and t, as in the i^receding case, is a function of x, x being given by the equation of partial differences, i-j— c — •^* )1~[ » Cdx J Cdt J in which z is any function whatever of x. In order to reduce u into a series proceeding according to the C d^u") powers of a, the value of } — -^\^ must be determined in the case in which ji=:0 ; but in consequence of the proposed equation of partial differences, we have \din _ d.fz.du ,^ (Jc) \- dt ' 134 CELESTIAL MECHANICS, fc5-terW~~'w*i dt r therefore, we shall have (dtci _ ldZ\~ This equation being differenced with respect to «, gives c d'u 1 _ d-.fz.du \'dJ'\~ do^.dt ' but the equation [k) gives, by changing u into fz.du, (.d.fz.du} _ ^ df^.du 1 id^VX dt y consequently [c?^M> d-.fz^.du ldA~ df ' This equation being differenced again with respect to a, gives id^Wi _ d^.Jz-.du w\~ dx.df ' but the equation (k) gives, by changing u into ^Vm . . du' du dx du' dx dii fz.du' * Let /.rf« =u', then -^ = -^. ^=--^-^ ^-rfT = '■ "1^- '^'^'^ '^ . d. fz.du _ d.Jz-.du substituting for du' its value, we obtain — ^j — . f As the characteristic/ indicates an operation, the reverse of that denoted byrf.we can remove the sign f, by depressing the index of d by unity. PART L— BOOK II. 135 therefore Ldx'\~ I df~y By continuing this process, it is easy to infer generally SdHi) U':fz\dzn (d''-\z\\-]\ id^^\=l—dF-r\ — r-^r Les us now suppose that by making «=0, we have x zz. T, T being a function t ; we shall substitute this value of x, in z, and in u. Let Z and u represent what these quantities then become ; we shall have on the hypothesis that azzO, ^d'tn di teS ~ dt^'~ ' and consequently, by what precedes, we shall obtain, - which gives . = u + «.Z._ + — .^.J^^+^^.c/^)^(+&c;(P) It only now remains to determine what function of t and «, x repre- sents ; which will be effected by the integration of the equation of partial differences jT-f^^'jjTf- For this purpose, we shall ob- serve, that rf^ = {§}.rf^4-{|}.^«: 136 CELESTIAL MECHANICS, and by substituting in place of < -r^t its value ^' ■< -7- r we will ob^ tain therefore, we shall have dz\ — -. <:f.(/+a3) 1 + \dxy Idt} which gives by its integration, x =^(t+a,z'), (i>(t-\-a.z) being an ar- bitrary function of t-\-o(,z; so that the quantity which we have termed T, is equal to qi^t). Consequently, as often as there exists between a. and X, an equation reducible to the form x = (p(^t+ az) ; the value of u will be determined by the formula (P) in a series proceeding according to the powers of «. dz * zd»'=dicz — a. —7-. dx, therefore, by substituting this value of zd», we obtain the dx expression for dx given in the text; now as dx is an exact differential, the member, at the dx right hand side of the equation must be also an exact differential, consequently, —r- •— I !-)-«. •—3—) > must be equal to (^(t-\-itz), ip' denoting the derivative function \ dx dt ' of (p. 2 being by hypothesis a function of x, let it equal F(x) and we shall have x = ^{t ■\-aF[x)), and it is easy to obtain from this expression the proposed differential equa- tion of partial differences, for dx_ da = ?'(<+«FW)- ^ (F(x))^»F'{x\^^ ^ -^ = «''('+«-fW) { ^+''-^<^)-^^ = and by eliminating <p'{'+«.F{x)), and reducing, we shall obtain dx , dx PART I.— BOOK II. 137 Let us now suppose, that m is a function of the two variables x and z', these variables being given by the equations of partial differences {d^\ -. fdx\ Cdx'\ _ , cdx'^ in which z and z' are any functions whatever of x and x'. It is easy to be assured that the integrals of these equations are respectively X = (p(^t-ir a.z) ; x' = v|/(/' + aV) ;* (p(f-)-«2), and tJ/C^'+ajV) being arbitrary functions, the one of t-^<xz, PART I. BOOK II. T and as u is supposed to be equal to <p(;r), du „ . dx du ,, , dx hence, by eliminating <p(i) we obtain —r- « -r- = —r- . -— -, and by substituting for — - da at dt dec da. its value F{x).-^—t and making Fix) =z, we obtain after ail reductions — ;— =z. — r— dt d» at ; dx d\x when 1 = ^ + »F{x) ; x=.t when « = 0; —-- =.1; a, Z, and —r- become respectively at. dt i|/(«), F(t), and -^'(t), consequently, the equation (P) will become 4'{t) + 4''{f)- P(*) «. d.{m).F{ty) u^ d\{^'(t). F(t)^) *3 „ .^ . ,. tion, azrl, then we shall have x = f-i-F(x), and the preceding series becomes ■^'(a:) == i|/< + r)-'(i). /■(<) + -— . ' ' •! — -^+ &c., which Lagrange first announced in 1772, an epoch deservedly celebrated in the history of science for the many beautiful applica- tions of this series, if F(x) = 1, then x= F{t-\-a), and ••• u =4'(''^)' * Let z=F(xx') ; 2' = F,{x x); :• x = <p(i + «. F(xy)), X = Mi" + «'• -f,(^ ^)), ■<" the functions indicated by F, F,, be defined, and if the form of the preceding equations permits us to eliminate, the values of z and z, may be respectively obtained in terms of X, t, a, t', we may v regard x, x', as functions of those four quantities. ^ =,'(*+*. F(x .')).(! +«.^). '£=,'(t+..Fix^HF+.. ^); 188 CELESTIAL MECHANICS, and the other of f+xz. Moreover, we have % =^(^+'-'-i^XV)Mi +<«'• ^);-|;= ^■(*' + «'.FXx/).«'. % ; § = 4^(.'+.'.F{x'x')).(l+.'. f ); % = ^'(^' + .'.F(xx')..'. ^' ; dx QX when a, a' vanish, we have — r- = <p'{t), -y- = <p'(t'). F, ; at a» dx ^ rfx „ dx' „ dx ,„ ,, daf ,,, ,, „ in this case x-=<p{t) ; x' = ■vf'(''), ••• !f is a function of t, t', only ; as u \t only an explicit function of x, x', we shall have du du dx du dz , , , , . , dx = — • U -r-,'-T- ; aid when « and « vanish — j- = (?'(;) F. aa dx da dx dot da dx' du du ,, . r, du du dx du dx' , , , „ — - =0; •.• -^ =— . (l>'(t). F; -— = -7-. — , + — . -J-;, and when a, a =0, rf« c?« rfar d* dx da dx da, dx daf . , » „ , du du , , , „ _ , , , dx , ,, . dx . . ^ = ^/^. F= *i. F; ^= 4^1, ^. F=^, F, .-. by substituting z for da dx dt dt ' da' d»'- df ' dt ' ^ F we obtain — — 2. —r- = 0 when x = <p{t + az), conversely, when this differential da at equation obtains, we can deternune the value of a; = cp(t-\-az). As a depends explicitly only on x, t', a', and as «' is one of the independent variables in differencing u with respect to a, it is only necessary to have respect to x, •.• the rea- soning of the preceding page is applicable in this case. du du . , . , . du When a is equal to cipher — = s. -jj-, •.• m this case we may substitute -^ - du for z. -^. PART I.— BOOK II. 139 This being premised, if we conceive that x is eliminated from u and from z, by means of the equation a/ = \{i! + a's') ; u and 2 will be- come functions of x, a! and t' without a or / ; therefore we shall ob- tain, by what goes before, I dx" y\ dt"-' y ' If we suppose «nO after the differentiations, and if besides, we make X = <p{t-\- a,z"") in the second member of this equation a; = (?(/+ ««"), and consequently i ;?- f = ^"^ -> ;7- f » ^^ shall have on these suppositions, and consequently, \d"'.u} \d«.\dx"''S ( dx S' dt" We shall have in like manner. m-{'-m\ df If we suppose a' to vanish after the differentiations, and if besides we suppose that in the second member of this equation, x' — »J/(/'+a'y»'); we shall obtain dr-\di" 140 CELESTIAL MECIHANCS, provided that we make a. and a' to vanish after the differentiations, and also that we suppose in the second member of this equation a; = ?.(^+«^") ; x' =^(lf-\-ai.'z"^)', which comes to supposing in the second member as well as in the first mem- and to change in the partial difference ] ;( , of this second c dx.da. J ber z into z", and z' into z''^. Thus, we shall have on those suppo- sitions, and also by changing z into Z, s/ into Z', and u into u, C d" -"'-"-. [ -^^^ } - __< l.da..da.') y ( 1.2.3 n. 1.2.3 n'.dr-\dt"^'-^ J ° By following on this reasoning, it is easy to infer, that if we have r equations, x"=n(r+o^"z")i &c. z, z', z", &c., being any functions whatever of x, x', sf', &c. j u being supposed to be a function of the same variables, we shall have generally t n+n'+n"+&c.-r^ C d'U ? ^ _-^ Xdx.da.' .d! a.' . &C.5 r" ^". »' « ' &c. - ( 1.2.3.. .w.l.2.3...?z'.1.2.3,..n".&c.c?r-'.rfr'-'.c?r.""-'' ^ c^-^M provided that in the partial difference < - — r? -y-v. o — f » ^^e change Ldx.da. .da. . &c. J z into s", e' into z'"', &c., and that afterwards we change z into Z, z' into Z', 2* into Z", &c., and m into u'. PART I.— BOOK II. 141 If there is but one variable x, we shall have CrfM? rdul therefore ■dt ^■(- 1^\ \ 9" = 1.2.3 n.dt"-'^ If there are two variables a: and x' ; we shall have this equation differenced with respect to «', gives but we have < — ; >■ = s'. j -7- f 5 ^^^ ^^ i" this equation x is sub- stituted in place of u, we have < — ? =2'. ^ ;7- c ; therefore S (Pu 1 _ i'^'^'idFsK , ^dz) idu) idZd:^^- -• I — irS ^^'\d'i;^'\di^' * By substituting 2" for z, &c. we have made the coefficient , ,, , gn,n to ae- pend on a coefficient of the second order, and the ilifFerentiations relative to t and t' will not be difficult when «, «' are = to cipher. du , du du'' , , , 4 du > <i d^i^ 1 , ^^ '^" -5— , = r. -7-., -.•8. , . , =Z,d.{z'. ■< -TT J- =Z2/. •< -; — 77 f + 2- — jT"* "X ' ■■• by substituting s", s'"', for s, s', respectively, we obtain the expression which is given in the text. 142 CELESTIAL MECHANICS, If we suppose « and «' equal to nothing, in the second member of this equation, and if we change 2 into Z", z into Z'"', and u into u ; we shall obtain the value of I -T — —J, on the same suppositions; hence we obtain 1 .S.-i n.dr-\ 1 .2.3 ri.dt''^-'- by proceeding in this manner the value of (/„, „.,„,„ &c., for any number of variables whatever, may be obtained. Although we have supposed that w, z, z, z'', &c., are functions of X, x, a/', &c., without t, t', f, &c. ; we can however suppose, that they contain these last variables : but then denoting these variables by tA t', t'', &c., it is necessary to suppose /, t', t'', constant in the differentiations, and after these operations to restore /, t', &c., in place of^,, //, &c. 22. Let us apply these results to the elliptic motion of the planets ; and for this purpose, let the equations (/") of No. 20, be resumed. The equation 7it — 11 — e. sin, u, or u ■=. nt + e. sin.«, being com- pared with X :=. 9(/ + az) ; x will be changed into u, t into nt, and a into e, z int